Question

Find the values of the three trigonometric ratios for angle A.

Short answer

The values of the three trigonometric ratios for angle A are $\sin \left(\angle A\right)=\frac{12}{13}$,$\cos \left(\angle A\right)=\frac{5}{13}$and $\tan \left(\angle A\right)=\frac{12}{5}$.

Step-by-step solution

Step 1. Given.

The values of the three trigonometric ratios for angle are given.

Step 2. Write the formulas for the trigonometric ratios.

The formula for the trigonometric ratios is:

$\sin \left(\angle A\right)=\frac{\text{legopposite}\angle \text{A}}{\text{hypotenuse}}$

$\cos \left(\angle A\right)=\frac{\text{legadjacentto}\angle \text{A}}{\text{hypotenuse}}$

$\tan \left(\angle A\right)=\frac{\text{legopposite}\angle \text{A}}{\text{legadjacentto}\angle \text{A}}$

Step 3. Find the values of the three trigonometric ratios for angle.

The given diagram is:

From the diagram, it can be noticed that the leg opposite $\angle A$, leg adjacent to $\angle A$, and the hypotenuse are 12, 5, and 13 respectively.

Therefore,

$$\begin{gathered} \sin \left(\angle A\right)=\frac{\text{legopposite}\angle \text{A}}{\text{hypotenuse}} \\ =\frac{12}{13} \end{gathered}$$

$$\begin{gathered} \cos \left(\angle A\right)=\frac{\text{legadjacentto}\angle \text{A}}{\text{hypotenuse}} \\ =\frac{5}{13} \end{gathered}$$

width="219" height="88" role="math">tan∠A=legopposite∠Alegadjacentto∠A=125

Therefore, the values of the three trigonometric ratios for angle A are $\sin \left(\angle A\right)=\frac{12}{13}$,$\cos \left(\angle A\right)=\frac{5}{13}$ and $\tan \left(\angle A\right)=\frac{12}{5}$.