PAS MATEMATIKA BLOG

Contoh Soal 6 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 26.&\textrm{Jika nilai}\: \: \cot A+\cos A=x\\ &\textrm{dan}\: \: \cot A-\cos A=y,\\ &\textrm{maka nilai}\: \: \left ( x^{2}-y^{2} \right )=\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&4\sqrt{xy}\\ \textrm{b}.&2\sqrt{xy}\\ \textrm{c}.&xy\\ \textrm{d}.&2\\ \textrm{e}.&4 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\color{magenta}xy&=\left (\cot A+\cos A \right )\left ( \cot A-\cos A \right )\\ &=\cot ^{2}A-\cos ^{2}A\\ &=\displaystyle \frac{\cos^{2}A }{\sin ^{2}A}-\cos ^{2}A\\ &=\displaystyle \frac{\cos^{2}A }{\sin ^{2}A}-\cos ^{2}A\times \frac{\sin ^{2}A}{\sin ^{2}A}\\ &=\displaystyle \frac{\cos ^{2}A}{\sin ^{2}A}\left ( 1-\sin ^{2}A \right )\\ &=\displaystyle \frac{\cos ^{4}A}{\sin ^{2}A}\\ \sqrt{xy}&=\displaystyle \frac{\cos A}{\sin A}\times \cos A\\ \color{black}\textrm{Se}&\color{black}\textrm{lanjutnya}\\ x^{2}-y^{2}&=\left (\cot A+\cos A \right )^{2}-\left ( \cot A-\cos A \right )^{2}\\ (x+y)&(x-y)=(\cot A+\cos A+\cot A-\cos A)\\ &\qquad\times (\cot A+\cos A-(\cot A-\cos A))\\ x^{2}-y^{2}&=2\cot A\times 2\cos A\\ &=4\times \displaystyle \frac{\cos A}{\sin A}\times \cos A\\ &=\color{magenta}4\sqrt{xy} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 27.&\textrm{Jika nilai}\: \: \cos A+\sin A=\sqrt{2}\cos A\\ &\textrm{maka nilai}\: \: \left ( \cos A-\sin A \right )=\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-\sqrt{2}\cos A\\ \textrm{b}.&-\sqrt{2}\sin A\\ \color{red}\textrm{c}.&\sqrt{2}\sin A\\ \textrm{d}.&\displaystyle \frac{1}{\sqrt{2}}\sec A\\ \textrm{e}.&\displaystyle \frac{1}{\sqrt{2}}\csc A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\cos A+\sin A&=\sqrt{2}\cos A\\ \left (\cos A+\sin A \right )^{2}&=\left (\sqrt{2}\cos A \right )^{2}\\ 1+2\sin A\cos A&=2\cos ^{2}A\\ 2\sin A\cos A&=2\cos ^{2}A-1\\ \textrm{maka}&\\ \left (\cos A-\sin A \right )^{2}&=1-2\sin A\cos A\\ &=1-\left ( 2\cos ^{2}A-1 \right )\\ &=2-2\cos ^{2}A\\ &=2\left ( 1-\cos ^{2}A \right )\\ &=2\sin ^{2}A\\ \cos A-\sin A&=\sqrt{2\sin ^{2}A}\\ &=\sin A\sqrt{2}\\ &=\color{magenta}\sqrt{2}.\sin A \end{aligned} \end{array}$

DAFTAR PUSTAKA

Sembiring, S., Zulkifli, M., Marsito, dan Rusdi, I. 2017. Matematika untuk Siswa SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Bandung: SEWU.
Sukino. 2016. Matematika untuk SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Jakarta: ERLANGGA

Contoh Soal 5 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 21.&\textrm{Nilai dari}\: \: \displaystyle \frac{1}{\sec ^{2}A}+\frac{1}{\csc ^{2}A}=....\\ &\begin{array}{llll}\\ \textrm{a}.&-\infty \\ \textrm{b}.&\displaystyle -1\\ \textrm{c}.&\displaystyle 0\\ \color{red}\textrm{d}.&\displaystyle 1\\ {e}.&\displaystyle \infty \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}&\displaystyle \frac{1}{\sec ^{2}A}+\frac{1}{\csc ^{2}A}\\ &=\cos ^{2}A+\sin ^{2}=1 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 22.&\textrm{Nilai dari}\: \: \displaystyle \frac{\tan B+\tan C}{\cot B+\cot C}=....\\ &\begin{array}{llll}\\ \textrm{a}.&\cot B\times \cot C \\ \color{red}\textrm{b}.&\displaystyle \tan B\times \tan C\\ \textrm{c}.&\displaystyle \sec B\times \csc C\\ \textrm{d}.&\displaystyle \tan B\times \cot C\\ {e}.&\displaystyle \tan B\times \csc C \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}&\displaystyle \frac{\tan B+\tan C}{\cot B+\cot C}\\ &=\displaystyle \frac{\tan B+\tan C}{\displaystyle \frac{1}{\tan B}+\frac{1}{\tan C}}\\ &=\displaystyle \frac{\tan B+\tan C}{\left ( \displaystyle \frac{\tan B+\tan C}{\tan B\times \tan C} \right )}\\ &=\tan B\times \tan C \end{aligned} \end{array}$

$\begin{array}{ll}\\ 23.&\textrm{Nilai dari}\\ &\displaystyle \frac{\tan A}{\sec A-1}+\frac{\tan A}{\sec A+1}=....\\ &\begin{array}{llll}\\ \textrm{a}.&2\tan A \\ \textrm{b}.&2\cot A\\ \textrm{c}.&\displaystyle 2\sec A\\ \color{red}\textrm{d}.&\displaystyle 2\csc A\\ {e}.&\displaystyle 2\tan A.\sec A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}&\displaystyle \frac{\tan A}{\sec A-1}+\frac{\tan A}{\sec A+1}\\ &=\tan A\left (\displaystyle \frac{1}{\displaystyle \frac{1}{\cos A}-1}+\frac{1}{\displaystyle \frac{1}{\cos A}+1} \right )\\ &=\displaystyle \frac{\sin A}{\cos A}\left ( \displaystyle \frac{\cos A}{1-\cos A}+\frac{\cos A}{1+\cos A} \right )\\ &=\displaystyle \frac{\sin A}{1-\cos A}+\frac{\sin A}{1+\cos A}\\ &=\displaystyle \frac{\sin A(1+\cos A)+\sin A(1-\cos A)}{(1-\cos A)(1+\cos A)}\\ &=\displaystyle \frac{2\sin A}{1-\cos ^{2}}\\ &=\displaystyle \frac{2\sin A}{\sin ^{2}A}\\ &=\displaystyle \frac{2}{\sin A}\\ &=2\csc A \end{aligned}\\\\ &\textrm{Sebagai catatanya}\\ &\textrm{Anda bisa gunakan cara yang lain} \end{array}$

$\begin{array}{ll}\\ 24.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &\sin \left ( 2x-20^{\circ} \right )=-\cos \left ( 3x+50^{\circ} \right )\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-30^{\circ}\\ \textrm{b}.&-25^{\circ}\\ \color{red}\textrm{c}.&20^{\circ}\\ \textrm{d}.&25^{\circ}\\ {e}.&30^{\circ} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\sin \left ( 2x-20^{\circ} \right )&=-\cos \left ( 3x+50^{\circ} \right )\\ \sin \left ( 20^{\circ}-2x \right )&=\cos \left ( 3x+50^{\circ} \right )\\ \sin A&=\cos B,\: \: \color{black}\textrm{artinya}\\ A+B&=90^{\circ},\: \: \color{magenta}\textrm{maka}\\ \left ( 20^{\circ}-2x \right )+\left ( 3x+50^{\circ} \right )&=90^{\circ}\\ x+70^{\circ}&=90^{\circ}\\ x&=90^{\circ}-70^{\circ}\\ &=20^{\circ} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 25.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &\tan \left ( 2x+60^{\circ} \right )=\cot \left ( 90^{\circ}-3x \right )\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&20^{\circ}\\ \textrm{b}.&30^{\circ}\\ \textrm{c}.&40^{\circ}\\ \textrm{d}.&50^{\circ}\\ \color{red}\textrm{e}.&60^{\circ} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}\tan \left ( 2x+60^{\circ} \right )&=\cot \left ( 90^{\circ}-3x \right )\\ \tan (2x+60^{\circ})&=\tan 3x\\ 2x+60^{\circ}&=3x\\ 2x-3x&=-60^{\circ}\\ -x&=-60^{\circ}\\ x&=60^{\circ} \end{aligned} \end{array}$

Contoh Soal 4 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 16.&\textrm{Jika}\: \: \tan^{2} x +\sec x =5 \: \: \textrm{untuk}\\ &0\leq x\leq \displaystyle \frac{\pi }{2}\: \: \textrm{maka nilai} \: \: \cos x=....\\ &\begin{array}{llll}\\ \textrm{a}.&0\\ \color{red}\textrm{b}.&\displaystyle \frac{1}{2}\\ \textrm{c}.&\displaystyle \frac{1}{3}\\ \textrm{d}.&\displaystyle \frac{1}{\sqrt{2}}\\ \textrm{e}.&\displaystyle \frac{1}{2}\sqrt{3} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\textrm{Ingat bahwa}&\: \: 0\leq x\leq \displaystyle \frac{\pi }{2}\\ \textrm{berarti sudu}&\textrm{t}\: \: x\: \: \textrm{berada di kuadran I}\\ \textrm{sehingga ak}&\textrm{an menyebabkan nilai}\\ & \color{magenta}\cos x=+\\ \color{black}\textrm{Selanjutnya}&\\ \tan^{2} x +\sec x &=5\\ \sec ^{2}x-1+\sec x&=5\\ \sec ^{2}x+\sec x-6&=0\\ (\sec x+3)(\sec x-2)&=0\\ \sec x=-3\: \: \textrm{atau}&\sec x=2\\ \textrm{untuk}\: \: \sec x&=-3\: \: (\color{red}\textrm{tidak memenuhi})\\ \textrm{untuk}\: \: \sec x&=2\: \: (\color{magenta}\textrm{memenuhi})\\ \color{black}\textrm{Selanjutnya}&\: \color{black}\textrm{lagi}\\ \sec x&=2\\ \displaystyle \frac{1}{\cos x}&=2\\ \cos x&=\color{magenta}\displaystyle \frac{1}{2} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 17.&\textrm{Nilai}\\ &\left ( \sin A+\cos A \right )^{2}+\left ( \sin A-\cos A \right )^{2}=....\\ &\begin{array}{llll}\\ \textrm{a}.&1\\ \color{red}\textrm{b}.&\displaystyle 2\\ \textrm{c}.&\displaystyle 3\\ \textrm{d}.&\displaystyle 3\cos A\\ {e}.&\displaystyle 4\sin A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}&\left ( \sin A+\cos A \right )^{2}+\left ( \sin A-\cos A \right )^{2}\\ &=\sin ^{2}A+2\sin A\cos A+\cos ^{2}A\\ &\quad +\sin ^{2}A-2\sin A\cos A+\cos ^{2}A\\ &=1+1\\ &=2 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 18.&\textrm{Nilai}\: \: \sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}}=....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\sec A+\tan A\\ \textrm{b}.&\sec ^{2}A+\tan ^{2}A\\ \textrm{c}.&\sec ^{2}A-\tan ^{2}A\\ \textrm{d}.&\tan ^{2}A-\sec ^{2}A\\ {e}.&\sec A\times \tan A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}&\sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}}\\ &=\sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}\times \frac{1+\sin A}{1+\sin A}}\\ &=\sqrt{\displaystyle \frac{\left ( 1+\sin A \right )^{2}}{1-\sin ^{2}A}}\\ &=\sqrt{\displaystyle \frac{\left ( 1+\sin A \right )^{2}}{\cos ^{2}A}}\\ &=\displaystyle \frac{1+\sin A}{\cos A}\\ &=\displaystyle \frac{1}{\cos A}+\frac{\sin A}{\cos A}\\ &=\sec A+\tan A \end{aligned} \end{array}$

$\begin{array}{ll}\\ 19.&\textrm{Jika}\: \: 0^{\circ}\leq \theta \leqslant 90^{\circ},\: \textrm{maka nilai}\\ &\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }-\frac{3+5\sin \theta }{4+5\cos \theta } \right )=....\\ &\begin{array}{llll}\\ \textrm{a}.&-1\\ \color{red}\textrm{b}.&\displaystyle 0\\ \textrm{c}.&\displaystyle \frac{1}{4}\\ \textrm{d}.&\displaystyle \frac{1}{2}\\ {e}.&\displaystyle 1 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}&\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }-\frac{3+5\sin \theta }{4+5\cos \theta } \right )\\ &=\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }\times \frac{4+5\cos \theta }{4+5\cos \theta } \right )\\ &\qquad-\left ( \displaystyle \frac{3+5\sin \theta }{4+5\cos \theta }\times \frac{3-5\sin \theta }{3-5\sin \theta } \right )\\ &=\displaystyle \frac{25\cos ^{2}-16-\left ( 9-25\sin ^{2}\theta \right )}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\displaystyle \frac{25\left ( \sin ^{2}\theta +\cos ^{2}\theta \right )-25}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\displaystyle \frac{0}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\color{magenta}0 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 20.&\textrm{Nilai}\\ &\left ( 1+\cot \theta -\csc \theta \right )\left ( 1+\tan \theta +\sec \theta \right )=....\\ &\begin{array}{llll}\\ \textrm{a}.&-2\\ \textrm{b}.&\displaystyle -1\\ \textrm{c}.&\displaystyle 0\\ \textrm{d}.&\displaystyle 1\\ \color{red}\textrm{e}.&\displaystyle 2 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}&\left ( 1+\cot \theta -\csc \theta \right )\left ( 1+\tan \theta +\sec \theta \right )\\ &=\left ( 1+\displaystyle \frac{\cos \theta }{\sin \theta } -\frac{1}{\sin \theta } \right )\left ( 1+\frac{\sin \theta }{\cos \theta } +\frac{1}{\cos \theta } \right )\\ &=\left ( \displaystyle \frac{\sin \theta +\cos \theta -1}{\sin \theta } \right )\left ( \displaystyle \frac{\cos \theta +\sin \theta +1}{\cos \theta } \right )\\ &=\displaystyle \frac{\left (\sin \theta +\cos \theta \right )^{2}-1}{\sin \theta \cos \theta }\\ &=\displaystyle \frac{1+2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }\\ &=2 \end{aligned} \end{array}$

Contoh Soal 3 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 11.&\textrm{Himpunan penyelesaian persamaan}\\ &3\cos 2x+5\sin x+1=0\: \: \textrm{untuk}\: \: 0^{\circ}\leq x\leq 2\pi \\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\left \{ \displaystyle \frac{7}{6}\pi ,\frac{11}{6}\pi \right \}\\ \textrm{b}.&\left \{ \displaystyle \frac{5}{6}\pi ,\frac{11}{6}\pi \right \}\\ \textrm{c}.&\left \{ \displaystyle \frac{1}{6}\pi ,\frac{7}{6}\pi \right \}\\ \textrm{d}.&\left \{ \displaystyle \frac{1}{5}\pi ,\frac{5}{6}\pi \right \}\\ \textrm{e}.&\left \{ \displaystyle \frac{5}{6}\pi ,\frac{7}{6}\pi \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}3\cos 2x+5\sin x+1&=0\\ 3\left ( 1-2\sin ^{2}x \right )+5\sin x+1&=0\\ -6\sin ^{2}+5\sin x+4&=0\\ 6\sin ^{2}x-5\sin x-4&=0\\ \left ( 3\sin x-4 \right )\left ( 2\sin x+1 \right )&=0\\ \sin x=\displaystyle \frac{4}{3}\: \: \textrm{atau}\: \: \sin x&=-\frac{1}{2}\\ \sin x&=\sin 150^{\circ}=\frac{5}{6}\pi \\ x&=\begin{cases} \displaystyle \frac{7}{6}\pi &+k.2\pi \\ \pi -\displaystyle \frac{7}{6}\pi & +k.2\pi \end{cases}\\ \textrm{saat}\: \: k&=0\\ x_{1}&=\displaystyle \frac{7}{6}\pi \\ x_{2}&=-\displaystyle \frac{1}{6}\pi \\ \textrm{saat}\: \: k&=1\\ x_{1}&=\color{red}\displaystyle \frac{7}{6}\pi +2\pi \\ x_{2}&=-\displaystyle \frac{1}{6}\pi +2\pi =\frac{11}{6}\pi \end{aligned} \end{array}$

$\begin{array}{ll}\\ 12.&\textrm{Himpunan penyelesaian dari}\\ &\sqrt{3}\sin 2x+2\cos ^{2}x=-1\: \: \textrm{untuk}\\ &0^{\circ}\leq x\leq 360^{\circ}\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\left \{ 240^{\circ},300^{\circ} \right \}\\ \textrm{b}.&\left \{ 30^{\circ},60^{\circ} \right \}\\ \textrm{c}.&\left \{ 150^{\circ},315^{\circ} \right \}\\ \color{red}\textrm{d}.&\left \{ 120^{\circ},300^{\circ} \right \}\\ \textrm{e}.&\left \{ 60^{\circ},150^{\circ} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\sqrt{3}\sin 2x+2\cos ^{2}x&=-1\\ \sqrt{3}\sin 2x+1+\cos 2x&=-1\\ \sqrt{3}\sin 2x+\cos 2x&=-2\\ \sqrt{\sqrt{3}^{2}+1^{2}}\cos \left ( 2x-\alpha \right )&=-2\\ a=x=1,\: \: b&=y=\sqrt{3}\\ \alpha &=\arctan \displaystyle \frac{b}{a}=\arctan \frac{\sqrt{3}}{1}\\ \alpha &=60^{\circ}\\ \textrm{maka persamaan akan}&\: \textrm{menjadi}\\ 2\cos \left ( 2x-60^{\circ} \right )&=-2\\ \cos \left ( 2x-60^{\circ} \right )&=-1\\ \cos \left ( 2x-60^{\circ} \right )&=\cos 180^{\circ}\\ \left ( 2x-60^{\circ} \right )&=\pm 180^{\circ}+k.360^{\circ}\\ 2x&=60^{\circ}\pm 180^{\circ}+k.360^{\circ}\\ x&=30^{\circ}\pm 90^{\circ}+k.180^{\circ}\\ \textrm{saat}\: \: k&=0\\ x_{1}&=120^{\circ}\\ x_{2}&=-60^{\circ}\\ \textrm{saat}\: \: k&=1\\ x_{3}&=120^{\circ}+360^{\circ}=....\\ x_{2}&=-60^{\circ}+360^{\circ}=300^{\circ} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 13.&\textrm{Jika}\: \: 3\sin \theta +4\cos \theta =5 \: \: \textrm{maka}\\ &\textrm{nilai dari} \: \: \sin \theta \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&0,3\\ \color{red}\textrm{b}.&0,60\\ \textrm{c}.&0,75\\ \textrm{d}.&0,80\\ \textrm{e}.&1,20 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}3\sin \theta +4\cos \theta &=5\\ k\cos \left ( \theta -\alpha \right )&=5\\ a=x=4,\: &b=y=3\\ \theta &=\arctan \displaystyle \frac{b}{a}\\ \theta &=\arctan \displaystyle \frac{3}{4}\\ \color{black}\textrm{atau}\: \: &\\ \tan \theta &=\displaystyle \frac{3}{4}\\ \color{black}\textrm{maka}\: \: &\\ \sin \theta &=\displaystyle \frac{3}{\sqrt{3^{2}+4^{2}}}\\ &=\displaystyle \frac{3}{5}\\ &=\color{magenta}0,6 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 14.&\textrm{Jika}\: \: \tan \theta +\sec \theta =x \: \: \textrm{maka}\\ &\textrm{nilai dari} \: \: \tan \theta \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle \frac{2x}{x^{2}-1}\\ \textrm{b}.&\displaystyle \frac{2x}{x^{2}+1}\\ \textrm{c}.&\displaystyle \frac{x^{2}+1}{2x}\\ \color{red}\textrm{d}.&\displaystyle \frac{x^{2}-1}{2x}\\ \textrm{e}.&\displaystyle \frac{x^{2}-1}{x^{2}+1} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\color{black}\textrm{Langkah}&\: \: 1\\ \tan \theta +\sec \theta &=x\\ \displaystyle \frac{\sin \theta }{\cos \theta } +\frac{1}{\cos \theta }&=x\\ \displaystyle \frac{\sin \theta +1}{\cos \theta }&=x\\ \sin \theta +1&=x\cos \theta ...........1\\ \color{black}\textrm{Langkah}&\: \: 2\\ \left (\tan \theta +\sec \theta \right )^{2}&=x^{2}\\ \tan ^{2}\theta +2\tan \theta \sec \theta +\sec ^{2}\theta &=x^{2}\\ \sec ^{2}\theta -1+2\tan \theta \sec \theta &+\sec ^{2}\theta =x^{2}\\ 2\sec ^{2}\theta +2\tan \theta \sec \theta &=x^{2}+1\\ \displaystyle \frac{2}{\cos ^{2}\theta }+\frac{2\sin \theta }{\cos ^{2}\theta }&=x^{2}+1\\ 1+\sin \theta &=\left (\displaystyle \frac{x^{2}+1}{2} \right )\cos ^{2}\theta.....2\\ \color{black}\textrm{Langkah}&\: \: 3\\ \left (\displaystyle \frac{x^{2}+1}{2} \right )\cos ^{2}&=x\cos \theta \\ \cos \theta &=\displaystyle \frac{2x}{x^{2}+1}\\ \textrm{maka}\: \: &(\color{magenta}\textbf{dengan sisi segitiga})\\ \tan \theta =\displaystyle \frac{\sqrt{\left ( x^{2}+1 \right )^{2}-(2x)^{2}}}{2x}&=\displaystyle \frac{\sqrt{x^{4}+2x^{2}+1-4x^{2}}}{2x}\\ \tan \theta &=\displaystyle \frac{\sqrt{x^{4}-2x^{2}+1}}{2x}\\ &=\displaystyle \frac{\sqrt{\left ( x^{2}-1 \right )^{2}}}{2x}\\ &=\displaystyle \frac{x^{2}-1}{2x} \end{aligned} \end{array}$

$\begin{aligned}.\: \: \qquad \textbf{Sehingga}&\: \textbf{dari kasus di atas didapatkan}\\ \color{magenta}\sin \theta &=\displaystyle \frac{x^{2}-1}{x^{2}+1}\\ \color{magenta}\cos \theta &=\displaystyle \frac{2x}{x^{2}+1}\\ \color{magenta}\tan \theta &=\displaystyle \frac{x^{2}-1}{2x} \end{aligned}$

$\begin{array}{ll}\\ 15.&\textrm{Jika}\: \: \sec x +\tan x =\displaystyle \frac{3}{2} \: \: \textrm{untuk}\\ &0\leq x\leq \displaystyle \frac{\pi }{2}\: \: \textrm{maka nilai} \: \: \sin x=....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\displaystyle \frac{5}{13}\\ \textrm{b}.&\displaystyle \frac{12}{13}\\ \textrm{c}.&\displaystyle 1\\ \textrm{d}.&\displaystyle \frac{2}{13}\\ \textrm{e}.&\displaystyle \frac{5}{12} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\sec x +\tan x &=\displaystyle \frac{3}{2}\\ \textrm{ingat saat}&\: \color{black}\textrm{mengerjakan soal no}.14\\ \sec \theta +\tan \theta &=x\\ \sin \theta &=\displaystyle \frac{x^{2}-1}{x^{2}+1},\: \: \color{black}\textrm{maka}\\ \sin x&=\color{magenta}\displaystyle \frac{\left ( \displaystyle \frac{3}{2} \right )^{2}-1}{\left (\displaystyle \frac{3}{2} \right )^{2}+1}\\ &=\displaystyle \frac{\displaystyle \frac{9}{4}-1}{\displaystyle \frac{9}{4}+1}\\ &=\frac{\displaystyle \frac{5}{4}}{\displaystyle \frac{13}{4}}\\ &=\color{magenta}\displaystyle \frac{5}{13} \end{aligned} \end{array}$

Contoh Soal 2 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 6.&\textrm{Bentuk}\: \: \sqrt{3}\cos x-\sin x\: \: \textrm{untuk}\: \: 0\leq x\leq 2\pi ,\\ &\textrm{dapat dinyatakan sebagai}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&2\cos \left ( x+\displaystyle \frac{\pi }{6} \right )\\ \textrm{b}.&2\cos \left ( x+\displaystyle \frac{7\pi }{6} \right ) \\ \color{red}\textrm{c}.&2\cos \left ( x-\displaystyle \frac{11\pi }{6} \right )\\ \textrm{d}.&2\cos \left ( x-\displaystyle \frac{7\pi }{6} \right )\\ \textrm{e}.&2\cos \left ( x-\displaystyle \frac{\pi }{6} \right ) \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textrm{c}\\ &\color{blue}\begin{aligned}\sqrt{3}\cos x-\sin x&=k\cos (x-\alpha )\\ (1)\qquad & (a,b)=\begin{cases} a &=\sqrt{3} \\ b &=-1 \end{cases}\\ \textrm{maka}\: &\textrm{titik ada dikadran IV}\\ (2)\: \quad k&=\sqrt{a^{2}+b^{2}}\\ &=\sqrt{\sqrt{3}^{2}+(-1)^{2}}\\ &=\sqrt{4}=2\\ (3)\quad \alpha &=\arctan \frac{b}{a}=\arctan \left (\frac{-1}{\sqrt{3}} \right )=-30^{\circ}\\ &=\left ( 360^{\circ}-30^{\circ} \right )=330^{\circ}=\displaystyle \frac{11}{6}\pi \\ \textrm{sehingga}&\\ \sqrt{3}\cos x-\sin x&=2\cos \left ( x-\displaystyle \frac{11}{6}\pi \right )\\ \end{aligned} \end{array}$

$\begin{array}{ll}\\ 7.&\textrm{Nilai-nilai}\: \: x\: \: \textrm{yang terletak pada}\: \: 0\leq x\leq 2\pi ,\\ &\textrm{yang memenuhi persamaan}\: \: \sqrt{3}\cos x+\sin x=\sqrt{2}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&75^{\circ}\: \: \textrm{atau}\: \: 285^{\circ}\\ \color{red}\textrm{b}.&75^{\circ}\: \: \textrm{atau}\: \: 345^{\circ}\\ \textrm{c}.&15^{\circ}\: \: \textrm{atau}\: \: 285^{\circ}\\ \textrm{d}.&15^{\circ}\: \: \textrm{atau}\: \: 345^{\circ}\\ \textrm{e}.&15^{\circ}\: \: \textrm{atau}\: \: 75^{\circ} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textrm{b}\\ &\color{blue}\begin{aligned}\sqrt{3}\cos x+\sin x&=\sqrt{2}\\ \sqrt{\sqrt{3}^{2}+1^{2}}&\left ( \cos \left ( \alpha -\arctan \displaystyle \frac{1}{\sqrt{3}} \right ) \right )=\sqrt{2}\\ 2\cos \left ( x-30^{\circ} \right )&=\sqrt{2}\\ \cos \left ( x-30^{\circ} \right )&=\displaystyle \frac{\sqrt{2}}{2}=\frac{1}{2}\sqrt{2}\\ \cos \left ( x-30^{\circ} \right )&=\cos 45^{\circ}\\ \left ( x-30^{\circ} \right )&=\pm 45^{\circ}+k.360^{\circ}\\ x&=30^{\circ}\pm 45^{\circ}+k.360^{\circ}\\ \textrm{saat}\: \: k&=0\\ x_{1}&=75^{\circ}\\ x_{2}&=\color{red}-15^{\circ}\\ \textrm{saat}\: \: k&=1\\ x_{3}&=75^{\circ}+360^{\circ}=\color{red}435^{\circ}\\ x_{4}&=-15^{\circ}+360^{\circ}=345^{\circ} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 8.&\textrm{Diketahui fungsi trigonometri}\: \: f(x)=\displaystyle \frac{1}{2}\sin 3x\\ &\textrm{perhatikanlah pernyataan-pernyataan berikut}\\ &(1)\quad \textrm{hasil dari}\: \: f(0)+f\left ( \displaystyle \frac{\pi }{6} \right )=1\\ &(2)\quad \textrm{hasil dari}\: \: f\left ( \displaystyle \frac{\pi }{6} \right )+f\left ( \displaystyle \frac{\pi }{3} \right )=\displaystyle \frac{1}{2}\\ &(3)\quad \textrm{hasil dari}\: \: f\left ( \displaystyle \frac{\pi }{6} \right )-f\left ( \displaystyle \frac{\pi }{3} \right )=\displaystyle \frac{1}{2}\\ &(4)\quad \textrm{hasil dari}\: \: f\left ( \displaystyle \frac{\pi }{3} \right )-f\left ( \displaystyle \frac{\pi }{6} \right )=1\\ &\textrm{Pernyataan yang tepat adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&(1)\: \: \textrm{dan}\: \: (2)\\ \textrm{b}.&(1)\: \: \textrm{dan}\: \: (3)\\ \textrm{c}.&(1)\: \: \textrm{dan}\: \: (4)\\ \color{red}\textrm{d}.&(2)\: \: \textrm{dan}\: \: (3)\\ \textrm{e}.&(3)\: \: \textrm{dan}\: \: (4) \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textrm{d}\\ &\color{blue}\begin{aligned}\textrm{Diketahui}&\\ f(x)&=\displaystyle \frac{1}{2}\sin 3x\\ \textrm{maka}\qquad&\\ f(0)&=\displaystyle \frac{1}{2}\sin 3(0^{\circ})=0\\ f\left ( \displaystyle \frac{\pi }{3} \right )&=\displaystyle \frac{1}{2}\sin 3\left ( \displaystyle \frac{\pi }{3} \right )=0\\ f\left ( \displaystyle \frac{\pi }{6} \right )&=\displaystyle \frac{1}{2}\sin 3\left ( \displaystyle \frac{\pi }{6} \right )=\displaystyle \frac{1}{2}\\ \end{aligned} \end{array}$

$\begin{array}{ll}\\ 9.&\textrm{Himpunan penyelesaian dari}\: \: \sin x-\sqrt{3}\cos x=-1\\ &\textrm{untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\left \{ 0^{\circ},120^{\circ} \right \}\\ \textrm{b}.&\left \{ 90^{\circ},330^{\circ} \right \}\\ \textrm{c}.&\left \{ 60^{\circ},180^{\circ} \right \}\\ \textrm{d}.&\left \{ 90^{\circ},120^{\circ} \right \}\\ \color{red}\textrm{e}.&\left \{ 30^{\circ},270^{\circ} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}\sin x-\sqrt{3}\cos x&=-1\\ \sqrt{1^{2}+\left ( -\sqrt{3} \right )^{2}}\cos \left ( x-\alpha \right )&=-1\\ a&=x=-\sqrt{3},\: \: b=y=1\\ &\textbf{kuadran II}\\ \alpha &=\arctan \left ( \displaystyle \frac{1}{-\sqrt{3}} \right )=-30^{\circ}\\ &=180^{\circ}-30^{\circ}=150^{\circ}\\ \textrm{maka persamaan akan}&\: \textrm{menjadi}\\ 2\cos \left ( x-150^{\circ} \right )&=-1\\ \cos \left ( x-150^{\circ} \right )&=\displaystyle \frac{-1}{2}\\ \cos \left ( x-150^{\circ} \right )&=\cos 120^{\circ}\\ \left ( x-150^{\circ} \right )&=\pm 120^{\circ}+k.360^{\circ}\\ x&=150^{\circ}\pm 120^{\circ}+k.360^{\circ}\\ \textrm{saat}\: \: &k=0\\ x_{1}&=270^{\circ}\\ x_{2}&=30^{\circ}\\ \textrm{saat}\: \: &k=1\\ x_{3}&=270^{\circ}+360^{\circ}=....\\ x_{4}&=30^{\circ}+360^{\circ}=.... \end{aligned} \end{array}$

$\begin{array}{ll}\\ 10.&\textrm{Nilai}\: \: \tan x\: \: \textrm{yang memenuhi persamaan}\\ &\cos 2x+7\cos x-3=0\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\sqrt{3}\\ \textrm{b}.&\displaystyle \frac{1}{2}\sqrt{3}\\ \textrm{c}.&\displaystyle \frac{1}{3}\sqrt{3}\\ \textrm{d}.&\displaystyle \frac{1}{2}\\ \textrm{e}.&\displaystyle \frac{1}{5}\sqrt{5} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\cos 2x+7\cos x-3&=0\\ 2\cos ^{2}x-1+7\cos x-3&=0\\ 2\cos ^{2}x+7\cos x-4&=0\\ \left ( \cos x+4 \right )\left (2 \cos -1 \right )&=0\\ \color{red}\cos x=-4\: \: \color{black}\textrm{atau}\: \: \color{magenta}\cos x&=\displaystyle \frac{1}{2}=\cos 60^{\circ}\\ x&=60^{\circ},\\ \textrm{maka}&\\ \tan 60^{\circ}&=\sqrt{3}\\\\ \textrm{ dan ingat bahwa}&\: \: \cos x=-4\: \: \color{red}\textrm{tidak memenuhi} \end{aligned} \end{array}$

Contoh Soal 1 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 1.&\textrm{Himpunan penyelesaian dari}\\ &\sin 2x=\displaystyle \frac{1}{2}\sqrt{3}\: \: \textrm{untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\left \{ 30^{\circ},210^{\circ} \right \}\\ \textrm{b}.&\left \{ 60^{\circ},240^{\circ} \right \}\\ \textrm{c}.&\left \{ 30^{\circ},60^{\circ},210^{\circ} \right \}\\ \color{red}\textrm{d}.&\left \{ 30^{\circ},60^{\circ},210^{\circ},240^{\circ} \right \}\\ \textrm{e}.&\left \{ 30^{\circ},60^{\circ},210^{\circ},240^{\circ},270^{\circ} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\sin 2x&=\displaystyle \frac{1}{2}\sqrt{3}\\ \sin 2x&=\sin 60^{\circ}\\ 2x&=\begin{cases} 60^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-60^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ x&=\begin{cases} 30^{\circ} & +k.180^{\circ}\\ 60^{\circ} & +k.180^{\circ} \end{cases}\\ \textrm{saat}&\: \: k=0\\ x&=\begin{cases} 30^{\circ} & \\ 60^{\circ} & \end{cases}\\ \textrm{saat}&\: \: k=1\\ x&=\begin{cases} 30^{\circ} & +1.180^{\circ}=210^{\circ}\\ 60^{\circ} & +1.180^{\circ}=240^{\circ} \end{cases}\\ \textrm{saat}&\: \: k=2\\ x&=\begin{cases} 30^{\circ} & +2.180^{\circ}=\color{red}390^{\circ}\\ 60^{\circ} & +2.180^{\circ}=\color{red}420^{\circ} \end{cases}\\ \color{red}\textrm{kedua}&\color{red}\textrm{nnya tidak memenuhi} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 2.&\textrm{Himpunan penyelesaian dari}\\ &\tan 2x-\sqrt{3}=0\: \: \textrm{untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\left \{ 15^{\circ},105^{\circ},195^{\circ},285^{\circ} \right \}\\ \color{red}\textrm{b}.&\left \{ 30^{\circ},120^{\circ},210^{\circ},300^{\circ} \right \}\\ \textrm{c}.&\left \{ 45^{\circ},135^{\circ},225^{\circ},315^{\circ} \right \}\\ \textrm{d}.&\left \{ 15^{\circ},105^{\circ},195^{\circ},285^{\circ} \right \}\\ \textrm{e}.&\left \{ 15^{\circ},30^{\circ},45^{\circ},60^{\circ},75^{\circ} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\tan 2x&-\sqrt{3}=0\\ \tan 2x&=\sqrt{3}\\ \tan 2x&=\tan 60^{\circ}\\ 2x&=60+k.180^{\circ}\\ x&=30^{\circ}+k.90^{\circ}\\ \textrm{saat}&\: \: k=0\\ x&=30^{\circ}\\ \textrm{saat}&\: \: k=1\\ x&=30^{\circ}+90^{\circ}=120^{\circ}\\ \textrm{saat}&\: \: k=2\\ x&=30^{\circ}+180^{\circ}=210^{\circ}\\ \textrm{saat}&\: \: k=3\\ x&=30^{\circ}+270^{\circ}=300^{\circ}\\ \textrm{saat}&\: \: k=4\\ x&=30^{\circ}+360^{\circ}=\color{red}390^{\circ}\\ \color{red}\textrm{tidak}&\: \color{red}\textrm{memenuhi} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 3.&\textrm{Himpunan penyelesaian dari}\\ &\cos 3x=-\displaystyle \frac{1}{2}\sqrt{3}\: \: \textrm{untuk}\: \: 0^{\circ}\leq x\leq 180^{\circ}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\left \{ 40^{\circ},80^{\circ} \right \}\\ \textrm{b}.&\left \{ 50^{\circ},70^{\circ} \right \}\\ \textrm{c}.&\left \{ 40^{\circ},70^{\circ},80^{\circ} \right \}\\ \color{red}\textrm{d}.&\left \{ 50^{\circ},70^{\circ},170^{\circ} \right \}\\ \textrm{e}.&\left \{ 50^{\circ},80^{\circ},170^{\circ} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\cos 3x&=-\displaystyle \frac{1}{2}\sqrt{3}\\ \cos 3x&=-\cos 30^{\circ}\\ \cos 3x&=\cos \left (180^{\circ}-30^{\circ} \right )=\cos 150^{\circ}\\ 3x&=\pm 150^{\circ}+k.360^{\circ}\\ x&=\pm 50^{\circ}+k.120^{\circ}\\ \textrm{saat}&\: \: k=0\\ x&=\pm 50^{\circ}\: \rightarrow x=50^{\circ}\: \: (\textrm{mm})\\ \textrm{saat}&\: \: k=1\\ x&=\pm 50^{\circ}+120^{\circ}=\begin{cases} 170^{\circ} & (\textrm{mm}) \\ 70^{\circ} & (\textrm{mm}) \end{cases} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 4.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &2\cos ^{2}x+\cos x-1=0\: \: \textrm{untuk}\: \: 0\leq x\leq \pi \\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\displaystyle \frac{1}{3}\pi \: \: \textrm{dan}\: \: \pi \\ \textrm{b}.&\displaystyle \frac{1}{3}\pi \: \: \textrm{dan}\: \: \frac{2}{3}\pi \\ \textrm{c}.&\displaystyle \frac{1}{3}\pi \: \: \textrm{dan}\: \: \frac{3}{4}\pi \\ \textrm{d}.&\displaystyle \frac{1}{4}\pi \: \: \textrm{dan}\: \: \frac{3}{4}\pi \\ \textrm{e}.&\displaystyle \frac{1}{4}\pi \: \: \textrm{dan}\: \: \frac{2}{3}\pi \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}2\cos ^{2}x+\cos x-1&=0\\ \left (2\cos x-1 \right )\left (\cos x+1 \right )&=0\\ \cos x=\displaystyle \frac{1}{2}\: \: \color{magenta}\textrm{atau}\: \: &\cos x=-1\\ \cos x=\cos 60^{\circ}=\cos \frac{1}{3}\pi \: \: &\\ \color{magenta}\textrm{atau}\: \: \cos x&=\cos 180^{\circ}=\cos \pi \\ \end{aligned} \end{array}$

$\begin{array}{ll}\\ 5.&\textrm{Untuk}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &\tan ^{2}x-\tan x-6=0\: \: \textrm{pada}\: \: 0\leq x\leq \pi ,\\ &\textrm{maka himpunan nilai}\: \: \sin x\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\left \{ \displaystyle \frac{3\sqrt{10}}{10},\frac{2\sqrt{5}}{5} \right \}\\ \textrm{b}.&\left \{ \displaystyle \frac{3\sqrt{10}}{10},-\frac{2\sqrt{5}}{5} \right \} \\ \textrm{c}.&\left \{ -\displaystyle \frac{3\sqrt{10}}{10},\frac{2\sqrt{5}}{5} \right \} \\ \textrm{d}.&\left \{ \displaystyle \frac{\sqrt{10}}{10},\frac{\sqrt{5}}{5} \right \} \\ \textrm{e}.&\left \{ \displaystyle \frac{\sqrt{10}}{10},\frac{2\sqrt{5}}{5} \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\tan ^{2}x-\tan x-6&=0\\ \left (\tan x-3 \right )\left (\tan x+2 \right )&=0\\ \tan x=3\: \: \textrm{atau}\: \: &\tan x=-2\\ \tan x=\displaystyle \frac{3}{1}\: \: \textrm{atau}\: \: &\tan x=\frac{-2}{1}\\ \sin x=\displaystyle \frac{3}{\sqrt{1^{2}+3^{2}}}\: \: \textrm{atau}\: \: &\sin x=\frac{2}{\sqrt{1^{2}+2^{2}}}\\ \sin x=\displaystyle \frac{3}{\sqrt{10}}\: \: \textrm{atau}\: \: &\sin x=\frac{2}{\sqrt{5}}\\ \sin x=\displaystyle \frac{3}{10}\sqrt{10}\: \: \textrm{atau}\: \: &\sin x=\frac{2}{5}\sqrt{5} \end{aligned} \end{array}$

Lanjutan 3 : Grafik Fungsi Trigonometri (Matematika Peminatan Kelas XI)

$\color{blue}\textrm{3. Grafik Fungsi Tangen}$

$\color{blue}\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \color{black}x&0&\frac{\pi }{6}&\frac{\pi }{4}&\frac{\pi }{3}&\frac{\pi }{2}&\frac{2\pi }{3}&\frac{3\pi }{4}&\frac{5\pi }{6}&\pi \\\hline \color{red}f(x)&0&\frac{1}{3}\sqrt{3}&1&\sqrt{3}&\infty &-\sqrt{3}&-1&-\frac{1}{3}\sqrt{3}&0\\\hline \color{black}x&\frac{7\pi }{6}&\frac{5\pi }{4}&\frac{4\pi }{3}&\frac{3\pi }{2}&\frac{5\pi }{3}&\frac{7\pi }{4}&\frac{11\pi }{6}&2\pi&\\\hline \color{red}f(x)&\frac{1}{3}\sqrt{3}&1&\sqrt{3}&\infty &-\sqrt{3}&-1&-\frac{1}{3}\sqrt{3}&0&\\\hline \end{array}$

Pada fungsi Tangen demikian juga nanti Cotangen ada beberapa nilai yang tida terdefinisi. Dalam fungsi Tangen fungsi, nilai fungsi yang tidak terdefini terdapat saat nilai $x=\displaystyle \frac{\pi }{2}=90^{\circ}$ dan $x=\displaystyle \frac{3\pi }{2}=270^{\circ}$. Sehingga pada saat posisi nilai itu, maka dibuatlah garis putus-putus pada grafik yang dan ditampakkan berupa grais vertikal yang selanjutnya garis vertikal itu disebut sebagai asimtot.

LIMIT FUNGSI ALJABAR

$\color{blue}\textrm{A. Pendahuluan}$

Mengingat kembali definisi limit yang telah dipelajari sebelumnya di kelas XI, yaitu limit fungsi aljabar $f(x)$ yang didefinisikan dengan:

$\begin{aligned}\underset{x\rightarrow a }{\textrm{lim}}\: f(x)=L&\: \: \textbf{adalah}\, \: \textrm{Jika}\: \: x\: \: \textrm{mendekati}\: \: a\\ &\textrm{dengan tidak sama dengan}\: \: a,\\ &\textrm{maka nilai}\: \: f(x)\: \: \textrm{mendekati}\: \: L \end{aligned}$.

Perhatikan definisi di atas istilah $x\: \: \textrm{mendekati}\: \: a$ dituliskan dengan simbol $(x\rightarrow a)$. Suatu nilai limit dianggap ada jika nilai $f(x)$ mendekati $a$ dari arah kiri sama dengan nilai $f(x)$ mendekati $a$ dari arah kanan dengan nilai yang sama misalnya $L$. Jika disimbolkan pernyataan ini menjadi berikut

$\begin{aligned}\underset{x\rightarrow \color{blue}a^{-} }{\textrm{lim}}\: f(x)=\underset{x\rightarrow \color{blue}a^{+} }{\textrm{lim}}\: f(x)=\underset{x\rightarrow \color{blue}a }{\textrm{lim}}\: f(x)=L&\: \: \ \end{aligned}$.

$\begin{aligned}\textrm{Perlu di}&\textrm{perhatikan bahwa didekati dari}\\ \bullet \: \: \textbf{kiri}\: &\textrm{disimbolkan dengan}\: \: \underset{x\rightarrow \color{blue}a^{-} }{\textrm{lim}}\: f(x),\: \: \textrm{dan}\\ \bullet \: \: \, \textbf{kan}&\textbf{an}\: \: \textrm{disimbolkan dengan}\: \: \underset{x\rightarrow \color{blue}a^{+} }{\textrm{lim}}\: f(x) \end{aligned}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah nilai limit dari}\: \: f(x)=\displaystyle \frac{x^{2}-4}{x-2}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Perhatikanlah ketika fungsi} \: \: \displaystyle \frac{x^{2}-4}{x-2}\\ &\textrm{di sekitar}\: \: x=2\: \: \textrm{sebagaimana dalam tabel}\\ &\textrm{berikut} \end{aligned}\end{array}$.

$\begin{aligned}.\qquad&\textrm{Jadi, nilai}\: \: \underset{x\rightarrow \color{blue}2}{\textrm{lim}} \: \displaystyle \frac{x^{2}-4}{x-2}=2 \: \: \textrm{atau dapat dikatakan}\\ &\textrm{nilai} \: \: \underset{x\rightarrow \color{blue}2}{\textrm{lim}} \: \displaystyle \frac{x^{2}-4}{x-2}\: \: \textbf{ada}\\ &\textrm{meskipun nilai substitusi langsung}\: \: x=2\: \: \textrm{yaitu}\\ &f(0)=\displaystyle \frac{0^{2}-0}{0-0}=\frac{0}{0}\: \: \textrm{berupa bentuk tak}\\ &\textrm{tentu. Berikut ilustrasinya} \end{aligned}$

$\begin{array}{ll}\\ 2.&\textrm{Selidikilah limit fungsi berikut apakah}\\ &\textrm{memiliki harga limit}\\ &\underset{x\rightarrow \color{red}5}{\textrm{lim}} \:f(x),\: \: \textrm{untuk}\: \: f(x)=\begin{cases} x &\textrm{saat}\: \: x<5 \\ 5-x &\textrm{saat}\: \: x\geq 5 \end{cases}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Perhatikanlah ketika didekati dari kiri}\\\ &\textrm{yaitu}\: \: x\rightarrow \color{red}5^{-},\: \: \color{black}\textrm{maka}\: \: \underset{x\rightarrow \color{red}5^{-}}{\textrm{lim}} \:f(x)=x\\ & \textrm{atau}\: \: \underset{x\rightarrow \color{red}5^{-}}{\textrm{lim}} \:f(x)=5\\ &\textrm{boleh juga dituliskan dengan}\\ &\underset{x\rightarrow \color{red}5^{-}}{\textrm{lim}} \:f(x)=x=5.\: \: \textrm{Sedangkan ketika}\\ &\textrm{didekati dari arah kanan yaitu}\: \: x\rightarrow \color{red}5^{+},\\ &\textrm{maka}\: \: \underset{x\rightarrow \color{red}5^{+}}{\textrm{lim}} \:f(x)=\underset{x\rightarrow \color{red}5^{+}}{\textrm{lim}} \: (5-x)=5-5=0.\\ &\textrm{Karena nilai}\: \: \underset{x\rightarrow \color{red}5^{-}}{\textrm{lim}} \:f(x)\neq \underset{x\rightarrow \color{red}5^{+}}{\textrm{lim}} \:f(x),\: \: \textrm{maka}\\ &\textrm{nilai atau harga}\: \: \underset{x\rightarrow \color{red}5}{\textrm{lim}} \:f(x)\: \: \textbf{tidak ada}\\ &\textrm{Berikut ilustrasi gambarnya} \end{aligned}\end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Selidikilah limit fungsi berikut apakah}\\ &\textrm{memiliki harga limit}\\ &\underset{x\rightarrow \color{red}0}{\textrm{lim}} \:f(x),\: \: \textrm{untuk}\: \: f(x)=\cos x\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Perhatikanlah ketika didekati dari kiri}\\\ &\textrm{yaitu}\: \: x\rightarrow \color{red}0^{-},\: \: \color{black}\textrm{maka}\: \: \underset{x\rightarrow \color{red}0^{-}}{\textrm{lim}} \: \cos x\\ &\begin{array}{|c|c|c|c|c|c|c|}\hline x&-0,5&-0,4&-0,3&-0,2&-0,1&0\\\hline \cos x&...&...&0,999986&0,999994&0,9999985&1\\\hline \end{array}\\ &\textrm{Sedangkan ketika}\\ &\textrm{didekati dari arah kanan yaitu}\: \: x\rightarrow \color{red}0^{+},\\ &\textrm{maka}\: \: \underset{x\rightarrow \color{red}0^{+}}{\textrm{lim}} \:\cos x\\ &\begin{array}{|c|c|c|c|c|c|c|}\hline x&0&0,1&0,2&0,3&0,4&0,5\\\hline \cos x&1&0,9999985&0,999994&0,999986&...&...\\\hline \end{array}\\ &\textrm{Karena nilai}\: \: \underset{x\rightarrow \color{red}0^{-}}{\textrm{lim}} \:f(x)= \underset{x\rightarrow \color{red}0^{+}}{\textrm{lim}} \:f(x)=1,\: \: \textrm{maka}\\ &\textrm{nilai}\: \: \underset{x\rightarrow \color{red}0}{\textrm{lim}} \:\cos x\: \: \textbf{ ada}\\ &\textrm{Berikut ilustrasi gambarnya} \end{aligned} \end{array}$.

$\color{blue}\textrm{B. Sifat-Sifat Limit Fungsi}$

$\begin{aligned}&\textrm{Misalkan}\: \: f\: \: \textrm{dan}\: \: g\: \: \textrm{adalah fungsi-fungsi yang}\\ &\textrm{mempunyai nilai limit di titik sekitar}\: \: x=a\\ &\textrm{atau}\: \: (x\rightarrow a)\: \: \textrm{dan}\: \: c\: \: \textrm{adalah suatu konstanta}\\ &\textrm{serta}\: \: n\: \: \textrm{adalah suatu bilangan bulat positif},\\ &\textrm{maka berlaku sifat-sifat berikut}:\\ &\begin{array}{ll}\\ 1.&\underset{x\rightarrow a }{\textrm{lim}}\: \displaystyle c=c\\ 2.&\underset{x\rightarrow a }{\textrm{lim}}\: \displaystyle x^{n}=a^{n}\\ 3.&\underset{x\rightarrow a }{\textrm{lim}}\: c.f(x)=c.\underset{x\rightarrow a }{\textrm{lim}}\: f(x)\\ 4.&\underset{x\rightarrow a}{\textrm{lim}}\: \left ( f(x)\pm g(x) \right )=\underset{x\rightarrow a }{\textrm{lim}}\: f(x)\pm \underset{x\rightarrow a }{\textrm{lim}}\: g(x)\\ 5.&\underset{x\rightarrow a }{\textrm{lim}}\: \left ( f(x)\times g(x) \right )=\underset{x\rightarrow a }{\textrm{lim}}\: f(x)\times \underset{x\rightarrow a }{\textrm{lim}}\: g(x)\\ 6.&\underset{x\rightarrow a }{\textrm{lim}}\: \displaystyle \frac{f(x)}{g(x)}=\displaystyle \frac{\underset{x\rightarrow a }{\textrm{lim}}\: f(x)}{\underset{x\rightarrow a }{\textrm{lim}}\: g(x)}\\ 7.&\underset{x\rightarrow a }{\textrm{lim}}\: \left (f(x) \right )^{n}= \left [\underset{x\rightarrow a }{\textrm{lim}}\: f(x)) \right ]^{n}\\ 8.&\underset{x\rightarrow a }{\textrm{lim}}\: \sqrt[n]{f(x)}=\sqrt[n]{\underset{x\rightarrow \infty}{\textrm{lim}}\: f(x)},\quad \textrm{dengan}\: \: \underset{x\rightarrow a }{\textrm{lim}}\: f(x)\geq 0\\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad \textrm{dan}\: \: n\: \: \textrm{genap} \end{array} \end{aligned}$

Lanjutan 2 : Grafik Fungsi Trigonometri (Matematika Peminatan Kelas XI)

$\color{blue}\textrm{2. Grafik Fungsi Cosinus}$

$\color{blue}\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \color{black}x&0&\frac{\pi }{6}&\frac{\pi }{4}&\frac{\pi }{3}&\frac{\pi }{2}&\frac{2\pi }{3}&\frac{3\pi }{4}&\frac{5\pi }{6}&\pi \\\hline \color{red}f(x)&1&\frac{1}{2}\sqrt{3}&\frac{1}{2}\sqrt{2}&\frac{1}{2}&0&-\frac{1}{2}&-\frac{1}{2}\sqrt{2}&-\frac{1}{2}\sqrt{3}&-1\\\hline \color{black}x&\frac{7\pi }{6}&\frac{5\pi }{4}&\frac{4\pi }{3}&\frac{3\pi }{2}&\frac{5\pi }{3}&\frac{7\pi }{4}&\frac{11\pi }{6}&2\pi&\\\hline \color{red}f(x)&-\frac{1}{2}\sqrt{3}&-\frac{1}{2}\sqrt{2}&-\frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}\sqrt{2}&\frac{1}{2}\sqrt{3}&1&\\\hline \end{array}$

Notasi Sigma Lanjutan Induksi Matematika (Matematika Wajib Kelas XI)

$\LARGE\color{blue}\textrm{A. Pendahuluan}$

Notasi sigma dari asalnya dari yaitu dari huruf yunani yang memiliki makna jumlah. Dalam matematika lambang notasi sigma $"\sum"$ selanjutnya akan menunjukkan penjumlahan yang teratur sehingga penulisan sebuah deret dari suatu bilangan yang berpola tertentu dapat disederhanakan lebih ringkas.

Sebagai ilustrasinya untuk deretny adalah sebagai berikut

$\begin{array}{ll}\\ \textrm{a}.\quad 1+2+3+4+5+\cdots +100\\ \textrm{b}.\quad 1+3+5+7+9+\cdots +199\\ \textrm{c}.\quad 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+\cdots +100^{2}\end{array}$

Dari bentuk deret di atas jika dimodelkan dengan notasi sigma maka bentuknya akan menjadi lebih sederhana, yaitu:

$\color{blue}\begin{aligned}&\sum_{i=1}^{n}a_{i}=\color{magenta}a_{1}+a_{2}+a_{3}+\cdots +a_{n}\\ &\textrm{Dibaca}:\: \: "\textrm{Jumlah}\: \textrm{dari}\: a_{i}\: \textrm{untuk}\: \: i\\ &\textrm{dari 1 sampai dengan}\: \: n" \: \: \textrm{dan}\: \: a_{i}\: \textrm{adalah suku ke}-i \end{aligned}$

Sehingga contoh ilustrasi deret di atas jika dinotasikan dengan notasi sigma menjadi

$\begin{array}{ll}\\ \textrm{a}.\quad 1+2+3+4+5+\cdots +100=\sum_{i=1}^{100}i\\\\ \textrm{b}.\quad 1+3+5+7+9+\cdots +199=\sum_{i=1}^{100}(2i-1)\\\\ \textrm{c}.\quad 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+\cdots +100^{2}=\sum_{i=1}^{100}i^{2}\end{array}$

$\LARGE\color{blue}\textrm{B. Sifat-Sifat Notasi Sigma}$

Misalkan diketahui $a_{k}$ dan $b_{k}$ adalah suku ke-k dan C adalah sebuah konstanta, maka

$\color{blue}\begin{array}{l}\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}C=nC\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}C.a_{k}=C\sum_{k=1}^{n}a_{k}\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}\left ( a_{k}+b_{k} \right )=\displaystyle \sum_{k=1}^{n}a_{k}+\sum_{k=1}^{n}b_{k}\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}\left ( a_{k}+b_{k} \right )^{2}=\displaystyle \sum_{k=1}^{n}a_{k}^{2}+2\sum_{k=1}^{n}a_{k}b_{k}+\sum_{k=1}^{n}b_{k}^{2}\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n-1}a_{k}+a_{n}\\ &\blacklozenge \quad \displaystyle \sum_{k=1}^{n}a_{k}=\sum_{k=1}^{m}a_{k}+\sum_{k=m+1}^{n}a_{k},\quad 1<m<n \end{array}$

$\LARGE\color{yellow}\fbox{CONTOH SOAL}$

$\begin{array}{ll}\\ 1.&\textrm{Uraikan jumlah berikut dengan lengkap}\\ &\begin{array}{lll}\\ &\textrm{a}.\quad \displaystyle \sum_{k=1}^{4}k&\textrm{f}.\quad \displaystyle \sum_{k=1}^{3}2^{k}\\ &\textrm{b}.\quad \displaystyle \sum_{k=1}^{4}(k-3)&\textrm{g}.\quad \displaystyle \sum_{k=1}^{3}\frac{1}{3^{k}}\\ &\textrm{c}.\quad \displaystyle \sum_{k=1}^{4}5k&\textrm{h}.\quad \displaystyle \sum_{k=1}^{5}\left ( k^{2}+1 \right )\\ &\textrm{d}.\quad \displaystyle \sum_{k=1}^{3}(4k+2)&\textrm{i}.\quad \displaystyle \sum_{k=1}^{4}5\left ( \frac{2}{3} \right )^{k}\\ &\textrm{e}.\quad \displaystyle \sum_{k=1}^{3}(2k+3)&\textrm{j}.\quad \displaystyle \sum_{k=1}^{5}\left ( k^{2}+2k-3 \right ) \end{array} \end{array}$

$\begin{aligned}&\textrm{Jawab}\\ &\color{blue}\begin{array}{ll}\\ 1.&\textrm{Perhatikanlah,}\\ &\begin{array}{lll}\\ &\textrm{a}.\quad \displaystyle \sum_{k=1}^{4}k=1+2+3+4=10\\ &\textrm{b}.\quad \displaystyle \sum_{k=1}^{4}(k-3)=(1-3)+(2-3)+(3-3)+(4-3)=-2.\\ &\textrm{c}.\quad \displaystyle \sum_{k=1}^{4}5k=5.1+5.2+5.3+5.4=50\\ &\textrm{d}.\quad \displaystyle \sum_{k=1}^{3}(4k+2)=(4.1+2)+(4.2+2)+(4.3+2)=30\\ &\textrm{e}.\quad \displaystyle \sum_{k=1}^{3}(2k+3)=(2.1 +3)+(2.2 +3)+(2.3 +3)=21 \\ &\textrm{f}.\quad \displaystyle \sum_{k=1}^{3}2^{k}=2^{1}+2^{2}+2^{3}+2^{4}=2+4+8+16=30\\ &\textrm{g}.\quad \displaystyle \sum_{k=1}^{3}\frac{1}{3^{k}}=\displaystyle \frac{1}{3^{1}}+\frac{1}{3^{2}}+\frac{1}{3^{3}}=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}=\displaystyle \frac{9+3+1}{27}=\frac{13}{27}\\ &\textrm{h}.\quad \displaystyle \sum_{k=1}^{5}\left ( k^{2}+1 \right )=\cdots +\cdots +\cdots +\cdots +\cdots \\ &\textrm{i}.\quad \displaystyle \sum_{k=1}^{4}5\left ( \frac{2}{3} \right )^{k}=\cdots +\cdots +\cdots +\cdots \\ &\textrm{j}.\quad \displaystyle \sum_{k=1}^{5}\left ( k^{2}+2k-3 \right )=\cdots +\cdots +\cdots +\cdots +\cdots \\ \end{array} \end{array} \end{aligned}$

$\begin{array}{ll}\\ 2.&\textrm{Nyatakanlah penjumlahan berikut dengan notasi sigma}\\ &\textrm{a}.\quad 2+4+8+16+32+64\\ &\textrm{b}.\quad 2+6+18+54+162\\ &\textrm{c}.\quad 15+24+35+48\\ &\textrm{d}.\quad \displaystyle \frac{2}{3}+\frac{4}{5}+\frac{8}{7}+\frac{16}{9}+\frac{32}{11}\\ &\textrm{e}.\quad ab+a^{2}b^{2}+a^{3}b^{3}+a^{4}b^{4} \end{array}$

$\color{blue}\begin{aligned}&\color{black}\textrm{Jawab}\\ &(\textrm{a})\quad 2+4+8+16+32+64=\displaystyle \sum_{k=1}^{6}2^{k}\\ &(\textrm{b})\quad 2+6+18+54+162=\displaystyle \sum_{k=1}^{5}2.3^{k-1}\\ &(\textrm{c})\quad 15+24+35+48=\displaystyle \sum_{k=1}^{4}\left ( k^{2}+6k+8 \right )\\ &(\textrm{d})\quad \displaystyle \frac{2}{3}+\frac{4}{5}+\frac{8}{7}+\frac{16}{9}+\frac{32}{11}=\displaystyle \sum_{k=1}^{5}\frac{2^{k}}{(2k+1)}\\ &(\textrm{e})\quad ab+a^{2}b^{2}+a^{3}b^{3}+a^{4}b^{4}=\displaystyle \sum_{k=1}^{4}(ab)^{k}\\ \end{aligned}$

$\begin{array}{ll}\\ 3.&\textrm{Dengan menggunakan kaidah notasi sigma},\\ &\textrm{tunjukkan bahwa}\\ &\textrm{a}.\quad \displaystyle \sum_{k=1}^{6}(2k+3)=2\sum_{k=1}^{6}k+18\\ &\textrm{b}.\quad \displaystyle \sum_{k=3}^{8}(k+3)=\sum_{k=1}^{6}k+30\\ &\textrm{c}.\quad \displaystyle \sum_{k=2}^{5}\left ( 2k^{2}+3k+3 \right )=2\sum_{k=1}^{4}k^{2}+7\sum_{k=1}^{4}k+32\\ &\textrm{d}.\quad \displaystyle \sum_{k=0}^{5}k^{2}=\sum_{k=1}^{6}k^{2}-2\sum_{k=1}^{6}k+6\\ &\textrm{e}.\quad \displaystyle \sum_{k=3}^{6}\left ( k^{2}+2k-3 \right )=\sum_{k=1}^{6}k^{2}+6\sum_{k=1}^{4}+20 \end{array}$

$\color{blue}\begin{aligned}&\color{black}\textrm{Jawab}\\ &(\textrm{a})\quad \displaystyle \sum_{k=1}^{6}(2k+3)=\sum_{k=1}^{6}2k+\sum_{k=1}^{6}3\\ &\: \qquad =\sum_{k=1}^{6}2k+6.3=2\sum_{k=1}^{6}k+18\\ &(\textrm{b})\quad \displaystyle \sum_{k=3}^{8}(k+3)=\sum_{k=3-2}^{8-2}\left ( (k+2)+3 \right )\\ &\: \qquad =\sum_{k=1}^{6}(k+5)=\sum_{k=1}^{6}k+\sum_{k=1}^{6}5=\sum_{k=1}^{6}k+6.5\\ &\: \qquad=\sum_{k=1}^{6}k+30\\ &(\textrm{c})\quad \displaystyle \sum_{k=2}^{5}\left ( 2k^{2}+3k+3 \right )\\ &\: \qquad=\sum_{k=2-1}^{5-1}\left ( 2(k+1)^{2}+3(k+1)+3 \right )=\cdots\\ &(\textrm{d})\quad \displaystyle\sum_{k=0}^{5}k^{2}=\sum_{k=0+1}^{5+1}(k-1)^{2}=\cdots\\ &(\textrm{e})\quad \displaystyle \sum_{k=3}^{6}\left ( k^{2}+2k-3 \right )=\cdots\\ \end{aligned}$

$\LARGE\color{green}\fbox{LATIHAN SOAL}$

$\begin{array}{ll}\\ .&\textrm{Buktikanlah bahwa}\\ &\textrm{a}.\quad \displaystyle \sum_{k=6}^{12}k^{2}=\sum_{k=1}^{7}k^{2}+10\sum_{k=1}^{7}k+175\\ &\textrm{b}.\quad \displaystyle \sum_{k=1}^{n}(3k-1)^{2}=9\sum_{k=1}^{n}k^{2}-6\sum_{k=1}^{n}k+n\\ &\textrm{c}.\quad \displaystyle \sum_{k=m}^{n}a_{k}=\sum_{k=m+p}^{n+p}a_{k-p}\\ &\textrm{d}.\quad \displaystyle \sum_{i=m}^{n}a_{i}=\sum_{i=1}^{n}a_{i}-\sum_{i=1}^{m-1}a_{i}\\ &\textrm{e}.\quad \displaystyle \sum_{k=1}^{n}a_{k}=\sum_{k=0}^{n-1}a_{k+1}=\sum_{k=2}^{n+2}a_{k-1}\\ &\textrm{f}.\quad \displaystyle \sum_{k=1}^{n-5}a_{k}=\sum_{k=1}^{n}a_{k}-\sum_{k=(n-5)+1}^{n}a_{k} \end{array}$

DAFTAR PUSTAKA

Kuntarti, Sulistiyono, Kurnianingsih, S. 2005. Matematika untuk SMA dan MA Kelas XII Program Ilmu Alam. Jakarta: Gelora Aksara Pratama.
Kuntarti, Sulistiyono, Kurnianingsih, S. 2007. Matematika SMA dan MA untuk Kelas XII Semester 2 Program IPA Standar Isi 2006. Jakarta: ESIS.

Lanjutan 1 : Grafik Fungsi Trigonometri (Matematika Peminatan Kelas XI)

$\LARGE\textrm{B. Grafik Fungsi Trigonometri}$

Dalam melukis grafik fungsi trigonometri nantinya yang ditampilkan adalah nilai-nilai untuk sudut istimewa saja. Dan selanjutnya besar sudutnya disajikan dalam derajat dan atau radian.

Selanjutnya perhatikanlah uraian berikut

$\color{blue}\textrm{1. Grafik Fungsi Sinus}$

$\color{blue}\begin{array}{|c|c|c|c|c|c|c|c|c|c| }\hline \color{magenta}x&0&\frac{\pi }{6}&\frac{\pi }{4}&\frac{\pi }{3}&\frac{\pi }{2}&\frac{2\pi }{3}&\frac{3\pi }{4}&\frac{5\pi }{6}&\pi \\\hline \color{red}f(x)&0&\frac{1}{2}&\frac{1}{2}\sqrt{2}&\frac{1}{2}\sqrt{3}&1&\frac{1}{2}\sqrt{3}&\frac{1}{2}\sqrt{2}&\frac{1}{2}&0\\\hline \color{magenta}x&\frac{7\pi }{6}&\frac{5\pi }{4}&\frac{4\pi }{3}&\frac{3\pi }{2}&\frac{5\pi }{3}&\frac{7\pi }{4}&\frac{11\pi }{6}&2\pi & \\\hline \color{red}f(x)&-\frac{1}{2}&-\frac{1}{2}\sqrt{2}&-\frac{1}{2}\sqrt{3}&-1&-\frac{1}{2}\sqrt{3}&-\frac{1}{2}\sqrt{2}&-\frac{1}{2}&0&\\\hline \end{array}$

Induksi Matematika (Kelas XI Matematika Wajib)

$\LARGE\textrm{A. Pendahuluan}$

Misalkan kita menjumlahkan 100 bilangan ganjil pertama (anggap saja sebagai penjumlahan suku pertama sampai suku ke seratus) yaitu : 1+3+5+...+199, maka untuk memudahkannya kita dapat menentukan cara menjumlahkan dengan atau menurut pola tertentu sebagaimana ilstrasi berikut ini

$\begin{aligned}1&=1^{2}=S_{1}\\ 1+3&=2^{2}=S_{2}\\ 1+3+5&=3^{2}=S_{3}\\ 1+3+5+7&=4^{2}=S_{4}\\ 1+3+5+7+9&=5^{2}=S_{5}\\ \vdots &\\ 1+3+5+7+9+\cdots +(2n-1)&=n^{2}=S_{n}\\ \vdots &\\ 1+3+5+7+9+\cdots +199&=100^{2}=S_{100} \end{aligned}$

$\LARGE\textrm{B. Induksi Matematika}$

Pola bilangan tertentu dalam matematika sebagaimana misal contoh di atas dapat ditarik suatu bentuk umum. Selanjutnya untuk membuktikan bahwa suatu bentuk umum dari sebuah pernyataan berlaku, kita dapat menggunakan Induksi Matematika ini. Tentunya semunya dari pernyataan tersebut harus memenuhi kriteria tertentu. Sehingga Induksi Matematika dapat juga disebutkan sebagai proses pembuktian pernyataan (teorema) dari kejadian-kejadian khusus yang berlaku untuk setiap bilangan asli.

Dalam pembuktian dengan Induksi Matematika, perhatikanlah beberapa langkah-langkah ini

Misalkan $\color{blue}P(n)$ adalah suatu pernyataan yang akan dibuktikan kebenarannya untuk semua bilangan asli $\color{blue}n$, maka

$\color{blue}\begin{aligned}&\color{black}\textbf{Langkahnya:}\\ &(1)\quad \textrm{buktikan}\: \: \color{black}P(1)\: \: \textrm{benar untuk}\: \: n=1\\ &\, \: \qquad \color{magenta}\textrm{Selanjutnya disebut Langkah Basis}\\ &(2)\quad \textrm{Asumsikan pernyataan berlaku untuk}\: \: n=k,\\ &\, \: \qquad \textrm{yaitu}\: \: P(k)\: \: \textrm{benar, dengan}\: \: k\in A,\\ &\, \: \qquad \textrm{maka untuk}\: \: \color{black}n=k+1\: \: \textrm{bahwa}\: \: P(k+1)\\ &\, \: \qquad \textrm{juga benar}\\ &\, \: \qquad \color{magenta}\textrm{Selanjutnya disebut Langkah Induksi}\\ &(3)\quad \textrm{Setelah langkah}\: \: (1)\: \: \textrm{dan}\: \: (2)\: \: \textrm{terpenuhi}\\ &\, \: \qquad \textrm{atau benar, maka dapat disimpulkan bahwa}\\ &\, \: \qquad \color{black}P(n)\: \: \textrm{benar untuk setiap}\: \: n\\ &\, \: \qquad \color{magenta}\textrm{Selanjutnya disebut Konkulsi}\\ \end{aligned}$

$\LARGE\color{blue}\fbox{CONTOH SOAL}$

$\begin{aligned}(1)\quad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\ &\color{red}1+2+3+...+n = \displaystyle \frac{n(n+1)}{2}\\\\ &\textbf{Bukti} \end{aligned}$

$\begin{aligned}.\: \qquad \textrm{Diket}&\textrm{ahui}\: \: P(n)\equiv 1+2+3+...+n=\displaystyle \frac{n(n+1)}{2}\\ (\textrm{a})\quad&\color{magenta}\textrm{Langkah basis}\\ &P(1)\: \: \textrm{benar, karena}\: \: 1=\displaystyle \frac{1(1+1)}{2}\\ (\textrm{b})\quad&\color{magenta}\textrm{Langkah induksi}\\ &\textrm{Misalkan rumus berlaku untuk}\: \: n=k,\: \: \textrm{yaitu}\\ &P(k)\equiv 1+2+3+...+k=\displaystyle \frac{k(k+1)}{2},\: \textrm{maka}\\ &\textrm{untuk}\: \: n=k+1:\\ &1+2+3+\cdots +(k)+(k+1)=\displaystyle \frac{(k+1)(k+2)}{2}\\ &\underset{\displaystyle \frac{k(k+1)}{2}}{\underbrace{1+2+3+\cdots +(k)}}+(k+1)=\displaystyle \frac{(k+1)(k+2)}{2}\\ &\displaystyle \frac{k(k+1)}{2}+(k+1)=\displaystyle \frac{(k+1)(k+2)}{2}\\ &\displaystyle (k+1)\left ( \frac{k}{2}+1 \right )=\displaystyle \frac{(k+1)(k+2)}{2}\\ &\displaystyle (k+1)\frac{k+2}{2} =\displaystyle \frac{(k+1)(k+2)}{2}\\ &\displaystyle \frac{(k+1)(k+2)}{2}=\displaystyle \frac{(k+1)(k+2)}{2}\equiv P(k+1)\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{benar untuk}\: \: n=k+1\\ (\textrm{c})\quad&\color{magenta}\textrm{Kesimpulan/Konklusi}\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{berlaku untuk}\: \: \forall \: n\in \mathbb{N}. \end{aligned}$

$\begin{aligned}(2)\quad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\ &\color{red}1+3+5+7+...+(2n-1)=n^{2}\\\\ &\textbf{Bukti} \end{aligned}$

$\begin{aligned}.\: \qquad \textrm{Diket}&\textrm{ahui}\: \: P(n)\equiv 1+3+5+...+(2n-1)=n^{2}\\ (\textrm{a})\quad&\color{magenta}\textrm{Langkah basis}\\ &P(1)\: \: \textrm{benar, karena}\: \: 2.1-1=1^{2}\Leftrightarrow 1=1\\ (\textrm{b})\quad&\color{magenta}\textrm{Langkah induksi}\\ &\textrm{Misalkan rumus berlaku untuk}\: \: n=k,\: \: \textrm{yaitu}\\ &P(k)\equiv 1+3+5+...+(2k-1)=k^{2},\: \textrm{maka}\\ &\textrm{untuk}\: \: n=k+1:\\ &1+3+5+\cdots +(2k-1)+(2(k+1)-1)=(k+1)^{2}\\ &\underset{\displaystyle k^{2}}{\underbrace{1+3+5+\cdots +(2k-1)}}+(2(k+1)-1)=(k+1)^{2}\\ &\qquad\qquad\quad k^{2}+\: \: \quad\quad\quad\quad\quad (2(k+1)-1)=(k+1)^{2}\\ &\qquad\qquad\quad k^{2}+\: \: \quad\quad\quad\quad\quad (2k+2-1)\: \: \: \: =(k+1)^{2}\\ &\qquad\qquad\quad k^{2}+2k+1 \quad\quad\quad\qquad\qquad\qquad=(k+1)^{2}\\ &\qquad\qquad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (k+1)^{2}=P(k+1)\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{benar untuk}\: \: n=k+1\\ (\textrm{c})\quad&\color{magenta}\textrm{Kesimpulan/Konklusi}\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{berlaku untuk}\: \: \forall \: n\in \mathbb{N}. \end{aligned}$

$\begin{aligned}(3)\quad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\ &\color{red}2n-3<2^{n-2}\\\\ &\textbf{Bukti} \end{aligned}$

$\begin{aligned}.\: \qquad \textrm{Diket}&\textrm{ahui}\: \: P(n)\equiv 2n-3<2^{n-2}\\ (\textrm{a})\quad&\color{magenta}\textrm{Langkah basis}\\ &P(1)\: \: \textrm{benar, karena}\: \: 2.1-3<2^{1-2}\\ &\textrm{demikian pula untuk}\: \: n=2\\ (\textrm{b})\quad&\color{magenta}\textrm{Langkah induksi}\\ &\textrm{Misalkan rumus berlaku untuk}\: \: n=k,\: \: \textrm{yaitu}\\ &P(k)\equiv (2k-3)<2^{k-2},\: \textrm{maka}\\ &\textrm{untuk}\: \: n=k+1:\\ &2(k+1)-3=2k+2-3=(2k-3)+2<2^{k-2}+2\\ &\textrm{sehingga}\\ &(2k-3)+2<2^{k-2}+2<2^{k-2}+2^{k-2},\quad \textrm{untuk}\quad k\geq 3\\ &(2k-3)+2<2.2^{k-2}\\ &(2k-3)+2<2^{(k+1)-2}\\ &\textrm{maka rumus berlaku untuk}\: \: P(k+1)\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{benar untuk}\: \: n=k+1\\ (\textrm{c})\quad&\color{magenta}\textrm{Kesimpulan/Konklusi}\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{berlaku untuk}\: \: \forall \: n\in \mathbb{N}. \end{aligned}$

$\begin{aligned}(4)\quad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\ &\color{red}(1+h)^{n}\geq 1+nh\\\\ &\textbf{Bukti} \end{aligned}$

$\begin{aligned}.\: \qquad \textrm{Diket}&\textrm{ahui}\: \: P(n)\equiv (1+h)^{n}\geq 1+nh\\ (\textrm{a})\quad&\color{magenta}\textrm{Langkah basis}\\ &P(1)\: \: \textrm{benar, karena}\: \: (1+h)^{1}\geq 1+1h\\ (\textrm{b})\quad&\color{magenta}\textrm{Langkah induksi}\\ &\textrm{Misalkan rumus berlaku untuk}\: \: n=k,\: \: \textrm{yaitu}\\ &P(k)\equiv (1+h)^{k}\geq 1+kh,\: \textrm{maka}\\ &\textrm{untuk}\: \: n=k+1:\\ &(1+h)^{k+1}\geq (1+kh)(1+h)\\ &(1+h)^{k+1}\geq \left ( 1+(k+1)h+kh^{2} \right )\\ &(1+h)^{k+1}\geq 1+(k+1)h\\ &\textrm{maka rumus berlaku untuk}\: \: P(k+1)\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{benar untuk}\: \: n=k+1\\ (\textrm{c})\quad&\color{magenta}\textrm{Kesimpulan/Konklusi}\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{berlaku untuk}\: \: \forall \: n\in \mathbb{N}. \end{aligned}$

$\color{blue}\begin{aligned}.\: \qquad &\textbf{Catatan}\\ &\textrm{untuk}\: \: k=2\\ &(1+h)^{2}=1+2h+h^{2}\geq 1+2h, \: \: \textrm{maka}\\ &(1+h)^{n}\geq 1+nh \end{aligned}$

$\begin{aligned}(5)\quad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\ &\color{red}\sum_{h=1}^{n}3^{h}=\displaystyle \frac{3}{2}\left ( 3^{n}-1 \right )\\\\ &\textbf{Bukti} \end{aligned}$

$\begin{aligned}.\: \qquad \textrm{Diket}&\textrm{ahui}\: \: P(n)\equiv \sum_{h=1}^{n}3^{h}=\displaystyle \frac{3}{2}\left ( 3^{n}-1 \right )\\ (\textrm{a})\quad&\color{magenta}\textrm{Langkah basis}\\ &P(1)\: \: \textrm{benar, karena untuk}\: \: n=1\\ &3^{1}=\displaystyle \frac{3}{2}\left ( 3^{1}-1 \right )\\ (\textrm{b})\quad&\color{magenta}\textrm{Langkah induksi}\\ &\textrm{Misalkan rumus berlaku untuk}\: \: n=k,\: \: \textrm{yaitu}\\ &P(k)\equiv \sum_{h=1}^{k}3^{h}=\displaystyle \frac{3}{2}\left ( 3^{k}-1 \right ),\: \textrm{maka}\\ &\textrm{untuk}\: \: n=k+1:\\ &\sum_{h=1}^{k+1}3^{h}=\sum_{h=1}^{k}3^{h}+\sum_{h=k+1}^{k+1}3^{h}\\ &=\displaystyle \frac{3}{2}\left ( 3^{k}-1 \right )+3^{k+1}\\ &=\displaystyle \frac{1}{2}\left ( 3^{k+1}-3+2.3^{k+1} \right )\\ &=\displaystyle \frac{1}{2}\left ( 3.3^{k+1}-3 \right )\\ &=\displaystyle \frac{3}{2}\left ( .3^{k+1}-1 \right )\\ &\textrm{maka rumus berlaku untuk}\: \: P(k+1)\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{benar untuk}\: \: n=k+1\\ (\textrm{c})\quad&\color{magenta}\textrm{Kesimpulan/Konklusi}\\ &\textrm{Jadi},\: \: P(n)\: \: \textrm{berlaku untuk}\: \: \forall \: n\in \mathbb{N}. \end{aligned}$

$\LARGE\color{blue}\fbox{LATIHAN SOAL}$

$\begin{aligned}.\: \qquad&\textrm{Buktikanlah dengan induksi matematika}\\ &\textrm{untuk}\: \: n\: \: \textrm{bilangan asli berlaku}\\\\ \end{aligned}$

$.\: \qquad\begin{array}{ll}\\ 1.&2+4+6+8+...+2n=n^{2}+n\\ 2.&1^{2}+2^{2}+3^{2}+...+n^{2}=\displaystyle \frac{1}{6}n(n+1)(2n+1)\\ 3.&1^{3}+2^{3}+3^{3}+...+n^{3}=\displaystyle \frac{1}{4}n^{2}(n+1)^{2}\\ 4.&1.2+2.3+3.4+...+n(n+1)=\displaystyle \frac{1}{3}n(n+1)(n+2)\\ 5.&\displaystyle \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{n(n+1)}=\displaystyle \frac{n}{n+1}\\ 6.&\displaystyle \frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{(2n-1)(2n+1)}=\displaystyle \frac{n}{2n+1}\\ 7.&\displaystyle \frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{n(n+1)(n+2)}=\displaystyle \frac{n(n+3)}{4(n+1)(n+2)}\\ 8.&1+\displaystyle \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n+1}-2\\ 9.&n^{3}-n\quad \textrm{habis dibagi oleh}\: \: 3\\ 10.&n^{5}-n\quad \textrm{habis dibagi oleh}\: \: 5\\ 11.&5^{n}+6.7^{n}+1\quad \textrm{habis dibagi oleh}\: \: 4\\ 12.&5^{2n}-1\quad \textrm{habis dibagi oleh}\: \: 3\\ 13.&3^{n}-1\geq 2^{n}\\ 14.&2n+7< (n+3)^{2}\\ 15.&2+4+6+8+...+2n\leq 2^{n}\\ 16.&\left ( 3+\sqrt{5} \right )^{n}+\left ( 3-\sqrt{5} \right )^{n}\quad \textrm{habis dibagi oleh}\: \: 2^{n} \end{array}$

DAFTAR PUSTAKA

Budhi, W.S., 2018. Bupena Mathematika SMA/MA Kelas XI Kelompok Wajib. Jakarta: Erlangga.
Kemendikbud. 2017. Matematika untuk SMA/MA/SMK Kelas XI Edisi Revisi. Jakarta: Kementerian Pendidikan Nasional.
Tampomas, H. 1999. SeribuPena Matematika Jilid 3 untuk SMU Kelas 3. Jakarta: Erlangga
Tim ITB. 2007. Program Pembinaan Kompetensi Siswa Bidang Matematika Tahap 1. Bandung: LPPM ITB

Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\LARGE\textrm{A. Persamaan Trigonometri}$

Ada minimal 3 yang utama untuk persmaan trigonometri sederhana, yaitu:

$\begin{aligned}1.\quad\sin x&=\sin \alpha \\ x&=\begin{cases} &=\alpha +k.360^{\circ} \\ &=\left ( 180^{\circ}-\alpha \right )+k.360^{\circ} \end{cases}\\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$.\: \quad \color{magenta}\textrm{atau}$

$\begin{aligned}.\: \quad\sin x&=\sin \alpha \\ x&=\begin{cases} &=\alpha +k.2\pi \\ &=\left ( \pi -\alpha \right )+k.2\pi \end{cases}\\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$\begin{aligned}2.\quad\cos x&=\cos \alpha \\ x&=\begin{cases} &=\alpha +k.360^{\circ} \\ &=-\alpha +k.360^{\circ} \end{cases}\\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$.\: \quad \color{magenta}\textrm{atau}$

$\begin{aligned}.\: \quad\cos x&=\cos \alpha \\ x&=\begin{cases} &=\alpha +k.2\pi \\ &=-\alpha +k.2\pi \end{cases}\\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$\begin{aligned}3.\quad\tan x&=\tan \alpha \\ x&=\alpha +k.180^{\circ}\\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$.\: \quad \color{magenta}\textrm{atau}$

$\begin{aligned}.\: \quad\tan x&=\tan \alpha \\ x&=\alpha +k.\pi \\ \textrm{deng}&\textrm{an}\\ k&\in \mathbb{Z} \end{aligned}$

$\color{yellow}\LARGE\fbox{CONTOH SOAL}$

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah himpunan penyelesaian dari }\\ &\textrm{persamaan-persamaan trigonometri berikut}\\ &\textrm{ini untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\\ &\begin{array}{lllllll}\\ \textrm{a}.& \sin x=\displaystyle \frac{1}{2}&\textrm{f}.& \tan x=-\displaystyle \frac{1}{3}\sqrt{3}&\textrm{k}.& \sin 2x=\displaystyle \frac{1}{2}\\ \textrm{b}.& \cos x=\displaystyle \frac{1}{2}\sqrt{3}&\textrm{g}& 2\cos x=-\sqrt{3}&\textrm{l}.& \cos 2x=-\displaystyle \frac{1}{2}\sqrt{3}\\ \textrm{c}.& \tan x=\sqrt{3}&\textrm{h}& 3\tan x=\sqrt{3}&\textrm{m}.& \tan 2x=\sqrt{3}\\ \textrm{d}.& \sin x=-1&\textrm{i}.& \sin x=\sin 46^{\circ}&\textrm{n}.& \sin \left ( 2x-30^{\circ} \right )=\sin 45^{\circ}\\ \textrm{e}.& \cos x=-\displaystyle \frac{1}{2}\sqrt{2}&\textrm{j}.& \cos x=\cos 93^{\circ}&\textrm{o}.& \sin \left ( 2x+60^{\circ} \right )=\sin 90^{\circ}\\ \end{array}\\ \end{array}$

$.\: \quad\color{blue}\textrm{Jawab}:$

$\color{blue}\begin{aligned}.\: \quad\textrm{a}.\quad\sin x&=\displaystyle \frac{1}{2}\\ \sin x&=\sin 30^{\circ}\\ x&=\begin{cases} 30^{\circ} & +k.360^{\circ}\\ \left (180^{\circ}-30^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 30^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ 150^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ k=1&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 30^{\circ},150^{\circ} \right \} \end{aligned}$

$\color{blue}\begin{aligned}.\: \quad\textrm{b}.\quad\cos x&=\displaystyle \frac{1}{2}\sqrt{3}\\ \cos x&=\cos 30^{\circ}\\ x&=\begin{cases} 30^{\circ} & +k.360^{\circ}\\ -30^{\circ} & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 30^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ -30^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \end{cases}\\ k=1&\\ x&=\begin{cases} 30^{\circ}+360^{\circ}=390^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \\ -30^{\circ}+360^{\circ}=330^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ \textrm{HP}&=\left \{ 30^{\circ},330^{\circ} \right \} \end{aligned}$

$\color{blue}\begin{aligned}.\: \quad\textrm{c}.\quad\tan x&=\sqrt{3}\\ \tan x&=\tan 60^{\circ}\\ x&=60^{\circ}+k.180^{\circ}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=60^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=1&\\ x&=60^{\circ}+180^{\circ}=240^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=2&\\ x&=60^{\circ}+360^{\circ}=420^{\circ}\: \: \color{red}\textrm{tidak memenuhi}\\ \textrm{HP}&=\left \{ 60^{\circ},240^{\circ} \right \} \end{aligned}$

$\color{blue}\begin{aligned}.\: \quad\textrm{d}.\quad\sin x&=-1\\ \sin x&= \sin 270^{\circ}\\ x&=\begin{cases} 270^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-270^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 270^{\circ} & \color{magenta}\textrm{memenuhi} \\ -90^{\circ} & \color{red}\textrm{tidak memenuhi} \end{cases}\\ k=1&\: \: \textrm{tidak memenuhi semuanya}\\ \textrm{HP}&=\left \{ 270^{\circ} \right \} \end{aligned}$

$\color{blue}\begin{aligned}.\: \quad\textrm{n}.\quad\sin \left ( 2x-30^{\circ} \right )&=\sin 45^{\circ}\\ \left ( 2x-30^{\circ} \right )&=\begin{cases} 45^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-45^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ 2x&=\begin{cases} 45^{\circ}+30^{\circ} &+k.360^{\circ} \\ 135^{\circ}+30^{\circ} &+ k.360^{\circ} \end{cases}\\ x&=\begin{cases} 37,5^{\circ} & +k.180^{\circ} \\ 82,5^{\circ} & +k.180^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 37,5^{\circ} & \\ 82,5^{\circ} & \end{cases}\\ k=1&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 37,5^{\circ}+180^{\circ} &=217,5^{\circ} \\ 82,5^{\circ}+180^{\circ} &=262,5^{\circ} \end{cases}\\ k=2&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 37,5^{\circ},82,5^{\circ},217,5^{\circ},262,5^{\circ} \right \} \end{aligned}$