# Perbandingan Trigonometri

### Segitiga siku-siku

##### 1. Segitiga Siku-siku

$$\sin \alpha = \dfrac yr = \dfrac {\text{depan}}{\text{miring}}$$

$$\cos \alpha = \dfrac xr = \dfrac {\text{samping}}{\text{miring}}$$

$$\tan \alpha = \dfrac yx = \dfrac {\text{depan}}{\text{samping}}$$

$$\sec \alpha = \dfrac {1}{\cos \alpha}$$

$$\csc \alpha = \dfrac {1}{\sin \alpha}$$

$$\cot \alpha = \dfrac {1}{\tan \alpha}$$

##### 2. Sudut Istimewa
 $$0$$ $$30^{\text{o}}$$ $$45^{\text{o}}$$ $$60^{\text{o}}$$ $$90^{\text{o}}$$ $$37^{\text{o}}$$ $$53^{\text{o}}$$ Sin 0 $$\dfrac{1}{2}$$ $$\dfrac{1}{2}\sqrt{2}$$ $$\dfrac{1}{2}\sqrt{3}$$ 1 $$0,6$$ $$0,8$$ Cos 1 $$\dfrac{1}{2}\sqrt{3}$$ $$\dfrac{1}{2}\sqrt{2}$$ $$\dfrac{1}{2}$$ 0 $$0,8$$ $$0,6$$ Tan 0 $$\dfrac{1}{3} \sqrt{3}$$ 1 $$\sqrt{3}$$ ∼ $$\dfrac 34$$ $$\dfrac 43$$

Contoh 01

Diketahui segitiga siku-siku di bawah ini:

Tentukan nilai dari $$\sin \alpha$$, $$\cos \alpha$$ dan $$\tan \alpha$$

Menentukan nilai r

\begin{equation*} \begin{split} x^2 + 5^2 & = r^2 \\\\ 3^2 + 4^2 & = r^2 \\\\ 9 + 16  & = r^2 \\\\ 25 & = r^2 \\\\ r & = \pm 5 \end{split} \end{equation*}

Gunakan nilai positif untuk $$r = 5$$

\begin{equation*} \sin \alpha = \frac {\text{depan}}{\text{miring}} = \bbox[5px, border: 2px solid magenta] {\frac {4}{5}} \end{equation*}

\begin{equation*} \cos \alpha = \frac {\text{samping}}{\text{miring}} = \bbox[5px, border: 2px solid magenta] {\frac {3}{5}} \end{equation*}

\begin{equation*} \tan \alpha = \frac {\text{depan}}{\text{samping}} = \bbox[5px, border: 2px solid magenta] {\frac {4}{3}} \end{equation*}

Contoh 02

Pada sebuah segitiga siku-siku ABC, diketahui nilai dari $$\sin A = \dfrac {5}{13}$$.

Tentukan nilai dari $$\cos A$$ dan $$\tan A$$

$$\sin A = \dfrac yr = \dfrac {\text{depan}}{\text{miring}} = \dfrac {5}{13}$$

Menentukan nilai $$x$$

\begin{equation*} \begin{split} x^2 + 5^2 & = 13^2 \\\\ x^2 + 25 & = 169 \\\\ x^2  & = 144 \\\\ x & = \pm 12 \end{split} \end{equation*}

Karena panjang AB harus bernilai positif, maka $$x = 12$$

$$\cos A = \dfrac xr = \dfrac {\text{samping}}{\text{miring}} = \bbox[5px, border: 2px solid magenta] {\dfrac {12}{13}}$$

$$\tan A = \dfrac yx = \dfrac {\text{depan}}{\text{samping}} = \bbox[5px, border: 2px solid magenta] {\dfrac {5}{12}}$$

##### SOAL LATIHAN

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