Identitas Trigonometri (Kelas XI MIPA)

Identitas Trigonometri adalah persamaan-persamaan yang berlaku untuk semua nilai pengganti peubah atau variabelnya yang di dalamnya terkandung perbandingan trigonometri.

\color{blue}\begin{array}{|c|c|}\hline Pythagoras&\begin{cases} \sin ^{2}\alpha +\cos ^{2}\alpha =1 \\ \sec ^{2}\alpha =\tan ^{2}\alpha +1 \\ \csc ^{2}\alpha =\cot ^{2}\alpha +1 \end{cases}\\\hline \textrm{Setengah}&\textrm{Satu}\\\hline \begin{cases} \sin \frac{1}{2}\alpha =\pm \sqrt{\displaystyle \frac{1-\cos \alpha }{2}} \\ \cos \frac{1}{2}\alpha =\pm \sqrt{\displaystyle \frac{1+\cos \alpha }{2}} \\ \tan \frac{1}{2}\alpha =\pm \sqrt{\displaystyle \frac{1-\cos \alpha }{1+\cos \alpha }} \end{cases}&\begin{cases} \sin \alpha =\displaystyle \frac{1}{\csc \alpha } \\ \cos \alpha =\displaystyle \frac{1}{\sec \alpha } \\ \tan \alpha =\displaystyle \frac{1}{\cot \alpha } \\ \tan \alpha =\displaystyle \frac{\sin \alpha }{\cos \alpha } \\ \cot \alpha =\displaystyle \frac{\cos \alpha }{\sin \alpha } \end{cases}\\\hline \textrm{Dua}&\textrm{Tiga}\\\hline \begin{cases} \sin 2\alpha =2\sin \alpha \cos \alpha \\ \cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha \\ \tan 2\alpha =\displaystyle \frac{2\tan \alpha }{1-\tan ^{2}\alpha } \end{cases}&\begin{cases} \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha \\ \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha \\ \tan 3\alpha =\displaystyle \frac{3\tan \alpha -\tan ^{3}\alpha }{1-3\tan ^{2}\alpha } \end{cases}\\\hline \end{array}

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

\begin{array}{ll}\\ 1.&\textrm{Dengan identitas}\: \: \sin ^{2}\gamma +\cos ^{2}\gamma =1,\: \textrm{buktikanlah identitas-identitas berikut}!\\ &\begin{array}{lllll}\\ \textrm{a}.&\left ( \sin \gamma -\cos \gamma \right )^{2}=1-\sin 2\gamma &\textrm{f}.&\displaystyle \frac{1+\sin \gamma }{\cos \gamma }+\displaystyle \frac{\cos \gamma }{1+\sin \gamma }=2\sec \gamma \\ \textrm{b}.&\displaystyle \frac{\csc ^{2}\gamma -1}{\csc ^{2}\gamma }=\cos ^{2}\gamma &\textrm{g}.&\displaystyle \frac{1}{1+\sin \gamma }+\displaystyle \frac{1}{1-\sin \gamma }=2\sec ^{2}\gamma \\ \textrm{c}.&\sqrt{\displaystyle \frac{1-\sin ^{2}\gamma }{1-\cos ^{2}\gamma }}=\cot \gamma &\textrm{h}.&\left ( \sec \gamma -\tan \gamma \right )^{2}=\displaystyle \frac{1-\sin \gamma }{1+\sin \gamma }\\ \textrm{d}.&\displaystyle \frac{1+\cos \gamma }{\sin ^{2}\gamma }=\displaystyle \frac{1}{1-\cos \gamma }&\textrm{i}.&\left ( \cot \gamma -\csc \gamma \right )^{2}=\displaystyle \frac{1-\cos \gamma }{1+\cos \gamma }\\ \textrm{e}.&\displaystyle \frac{1+\cos \gamma }{\sin \gamma }+\displaystyle \frac{\sin \gamma }{1+\cos \gamma }=2\csc \gamma &\textrm{j}.&\displaystyle \frac{\sin \gamma -2\sin ^{3}\gamma }{2\cos ^{3}\gamma -\cos \gamma }=\tan \gamma \end{array} \end{array}

Bukti:

\color{blue}\begin{aligned}1.\textrm{a}.\quad\left ( \sin \gamma -\cos \gamma \right )^{2}&=\left ( \sin \gamma -\cos \gamma \right )\times \left ( \sin \gamma -\cos \gamma \right )\\ &=\sin ^{2}\gamma -2\sin \gamma \cos \gamma +\cos ^{2}\gamma \\ &=\sin ^{2}\gamma +\cos ^{2}\gamma -2\sin \gamma \cos \gamma \\ &=1-\sin 2\gamma \qquad \color{black}\blacksquare \end{aligned}.

\begin{aligned}1.\textrm{b}.\quad\displaystyle \frac{\csc ^{2}\gamma -1}{\csc ^{2}\gamma }&=1-\displaystyle \frac{1}{csc^{2}\gamma }\\ &=1-\sin ^{2}\gamma \\ &=\cos ^{2}\gamma \qquad \blacksquare \end{aligned}.

\begin{aligned}1.\textrm{c}.\quad\sqrt{\displaystyle \frac{1-\sin ^{2}\gamma }{1-\cos ^{2}\gamma }}&=\sqrt{\displaystyle \frac{\cos ^{2}\gamma }{\sin ^{2}\gamma }}\\ &=\sqrt{\cot ^{2}\gamma }\\ &=\cot \gamma \qquad \blacksquare \end{aligned}.

\color{blue}\begin{aligned}1.\textrm{j}.\quad\displaystyle \frac{\sin \gamma -2\sin ^{3}\gamma }{2\cos ^{3}\gamma -\cos \gamma }&=\displaystyle \frac{\sin \gamma \left ( 1-2\sin ^{2}\gamma \right )}{\cos \gamma \left ( 2\cos ^{2}\gamma -1 \right )}\\ &=\tan \gamma \times \displaystyle \frac{\cos 2 \gamma }{\cos 2\gamma }\\ &=\tan \gamma \qquad \color{black}\blacksquare \end{aligned}

\begin{array}{ll}\\ 2.&\textrm{Dengan identitas}\: \: 1+\tan ^{2}\beta=\sec ^{2}\beta ,\: \textrm{buktikanlah identitas-identitas berikut}!\\ &\begin{array}{lllll}\\ \textrm{a}.&\displaystyle \frac{1+\tan ^{2}\beta }{\csc ^{2}\beta }=\tan ^{2}\beta &\textrm{f}.&1+\tan ^{2}\beta =\displaystyle \frac{\tan ^{2}\beta }{\sin ^{2}\beta }\\ \textrm{b}.&\displaystyle \frac{1}{\sec \beta +\tan \beta }=\sec \beta -\tan \beta &\textrm{g}.&\displaystyle \frac{\tan ^{2}\beta }{1+\tan ^{2}\beta }+\displaystyle \frac{\cot ^{2}\beta }{1+\cot ^{2}\beta }=1 \\ \textrm{c}.&\displaystyle \frac{1-\tan ^{2}\beta }{1+\tan ^{2}\beta }=2\cos ^{2}\beta -1&\textrm{h}.&\displaystyle \frac{1}{\sec \beta +\tan \beta }+\displaystyle \frac{1}{\sec \beta -\tan \beta }=2\sec \beta \\ \textrm{d}.&\displaystyle \frac{\sec \beta +\tan \beta }{\sec \beta -\tan \beta }=\left ( \sec \beta +\tan \beta \right )^{2}&\textrm{i}.&\left ( 1+\tan ^{2}\beta \right )\left ( 1+\displaystyle \frac{1}{\tan ^{2}\beta } \right )=\sec ^{2}\beta \csc ^{2}\beta\\ \textrm{e}.&\displaystyle \frac{\sec \beta -\tan \beta }{\sec \beta +\tan \beta }=\displaystyle \frac{\cos ^{2}\beta }{\left ( 1+\sin \beta \right )^{2}}&\textrm{j}.&\displaystyle \frac{\sec \beta -\tan \beta }{\sec \beta +\tan \beta }=1-2\sec \beta \tan \beta \end{array} \end{array}.

Bukti:

\color{blue}\begin{aligned}2.\textrm{f}.\quad 1+\tan ^{2}\beta &=\sec ^{2}\beta \\ &=\displaystyle \frac{1}{\cos ^{2}\beta }\\ &=\displaystyle \frac{1}{\cos ^{2}\beta }\times \displaystyle \frac{\sin ^{2}\beta }{\sin ^{2}\beta }\\ &=\displaystyle \frac{\sin ^{2}\beta }{\cos ^{2}\beta }\times \displaystyle \frac{1}{\sin ^{2}\beta }\\ &=\tan ^{2}\beta \times \displaystyle \frac{1}{\sin ^{2}\beta }\\ &=\displaystyle \frac{\tan ^{2}\beta }{\sin ^{2}\beta } \qquad \color{black}\blacksquare \end{aligned}.

\begin{array}{ll}\\ 3.&\textrm{Dengan identitas}\: \: 1+\cot ^{2}\alpha =\csc ^{2}\alpha ,\: \textrm{buktikanlah identitas-identitas berikut}!\\ &\begin{array}{lllll}\\ \textrm{a}.&\left ( \csc \alpha -\cot \alpha \right )^{2}=\displaystyle \frac{1-\cos \alpha }{1+\cos \alpha }&\textrm{f}.&\displaystyle \frac{\tan \alpha }{1-\cot \alpha }+\displaystyle \frac{\cot \alpha }{1-\tan \alpha }=1+\tan \alpha +\cot \alpha \\ \textrm{b}.& \sec ^{2}\alpha -\csc ^{2}\alpha =\tan ^{2}\alpha -\cot ^{2}\alpha &\textrm{g}.&1+\displaystyle \frac{\cot ^{2}\alpha }{1+\csc \alpha }=\csc \alpha \\ \textrm{c}.&\sqrt{\sec ^{2}\alpha +\csc ^{2}\alpha }=\tan \alpha +\cot \alpha &\textrm{h}.&\displaystyle \frac{\tan \alpha }{\left ( 1+\tan ^{2}\alpha \right )^{2}}+\displaystyle \frac{\cot \alpha }{\left ( 1+\cot ^{2}\alpha \right )^{2}}=\sin \alpha \cos \alpha \\ \textrm{d}.&\displaystyle \tan ^{2}\alpha +\cot ^{2}\alpha +2=\sec ^{2}\alpha \csc ^{2}\alpha &\textrm{i}.&\left ( \sec ^{2}\alpha -1 \right )\left ( \csc ^{2}\alpha -1 \right )=1\\ \textrm{e}.&\displaystyle \frac{1}{\csc \alpha -\cot \alpha }+\displaystyle \frac{1}{\csc \alpha +\cot \alpha }=2\csc \alpha &\textrm{j}.&\cot ^{2}\alpha \left ( \displaystyle \frac{\sec \alpha -1}{1+\sin \alpha } \right )+\sec ^{2}\alpha \left ( \displaystyle \frac{\sin \alpha -1}{1+\sec \alpha } \right )=0 \end{array} \end{array}.

Bukti:

\color{blue}\begin{aligned}3&.\textrm{j}.\quad \cot ^{2}\alpha \left ( \displaystyle \frac{\sec \alpha -1}{1+\sin \alpha } \right )+\sec ^{2}\alpha \left ( \displaystyle \frac{\sin \alpha -1}{1+\sec \alpha } \right )\\ &=\left ( \csc ^{2}\alpha -1 \right ) \left ( \displaystyle \frac{\sec \alpha -1}{1+\sin \alpha } \right )+\sec ^{2}\alpha \left ( \displaystyle \frac{\sin \alpha -1}{1+\sec \alpha } \right )\\ &=\left ( \displaystyle \frac{1}{\sin ^{2}\alpha } -1 \right ) \left ( \displaystyle \frac{\displaystyle \frac{1}{\cos \alpha } -1}{1+\sin \alpha } \right )+\displaystyle \frac{1}{\cos ^{2}\alpha } \left ( \displaystyle \frac{\sin \alpha -1}{1+\displaystyle \frac{1}{\cos \alpha } } \right )\\ &=\left ( \displaystyle \frac{1-\sin ^{2}\alpha }{\sin ^{2}\alpha } \right ) \left ( \displaystyle \frac{\displaystyle \frac{1-\cos \alpha }{\cos \alpha }}{1+\sin \alpha } \right )+\displaystyle \frac{1}{\cos ^{2}\alpha } \left ( \displaystyle \frac{\sin \alpha -1}{\displaystyle \frac{\cos \alpha +1}{\cos \alpha } } \right )\\ \end{aligned}.
\color{blue}\begin{aligned} .\: &=\left ( \displaystyle \frac{\cos ^{2}\alpha }{\sin ^{2}\alpha .\cos \alpha } \right ) \left ( \displaystyle \frac{1-\cos \alpha }{1+\sin \alpha } \right )+\displaystyle \frac{\cos \alpha }{\cos ^{2}\alpha } \left ( \displaystyle \frac{\sin \alpha -1}{\cos \alpha +1} \right )\\ .\: &=\left ( \displaystyle \frac{\cos \alpha }{\sin ^{2}\alpha } \right ) \left ( \displaystyle \frac{1-\cos \alpha }{1+\sin \alpha } \right )\times \left ( \displaystyle \frac{1-\sin \alpha }{1-\sin \alpha } \right )+\displaystyle \frac{1 }{\cos \alpha } \left ( \displaystyle \frac{\sin \alpha -1}{\cos \alpha +1} \right )\times \left ( \displaystyle \frac{\cos \alpha -1}{\cos \alpha -1} \right )\\ .\: &=\left ( \displaystyle \frac{\cos \alpha }{\sin ^{2}\alpha } \right ) \left ( \displaystyle \frac{1-\sin \alpha -\cos \alpha +\cos \alpha \sin \alpha }{1-\sin^{2} \alpha } \right )+\displaystyle \frac{1 }{\cos \alpha } \left ( \displaystyle \frac{\sin \alpha \cos \alpha -\sin \alpha -\cos \alpha +1}{\cos ^{2}\alpha -1} \right )\\ \end{aligned}.
\color{blue}\begin{aligned} .\: &=\left ( \displaystyle \frac{\cos \alpha }{\sin ^{2}\alpha } \right ) \left ( \displaystyle \frac{1-\sin \alpha -\cos \alpha +\cos \alpha \sin \alpha }{\cos^{2} \alpha } \right )+\displaystyle \frac{1 }{\cos \alpha } \left ( \displaystyle \frac{\sin \alpha \cos \alpha -\sin \alpha -\cos \alpha +1}{-\sin ^{2}\alpha } \right )\\ &=\left ( \displaystyle \frac{1}{\sin ^{2}\alpha \cos \alpha }-\displaystyle \frac{1}{\sin ^{2}\alpha \cos \alpha } \right ) \left ( 1-\sin \alpha -\cos \alpha +\cos \alpha \sin \alpha \right )\\ .\: &=0\qquad \color{black}\blacksquare \end{aligned}.

\colorbox{magenta}{\LARGE{\fbox{\LARGE{\fbox{LATIHAN SOAL}}}}}.

Untuk soal-soal yang belum ditunjukkan buktinya, silahkan gunakan sebagai latihan mandiri

Sumber Referensi

  1. Kurnia, N.,  dkk. 2017. Jelajah Matematika 2 SMA Kelas XI Peminatan MIPA(Edisi Revisi 2016). Jakarta: Yudistira.
  2. Tampomas, H. 1999. Seribu Pena Matematika SMU Jilid 1 Kelas 1. Jakarta: ERLANGGA.

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