Q65. - Algebra 1

Question

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

Step-by-Step Solution

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 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:

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

$\cos \left(\angle A\right)=\frac{\text{leg adjacent to }\angle \text{A}}{\text{hypotenuse}}$

$\tan \left(\angle A\right)=\frac{\text{leg opposite }\angle \text{A}}{\text{leg adjacent to }\angle \text{A}}$

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,

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

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

width="219" height="88" style="max-width: none; vertical-align: -45px;" $$\begin{gathered} \tan \left(\angle A\right)=\frac{\text{leg opposite }\angle \text{A}}{\text{leg adjacent to }\angle \text{A}} \\ =\frac{12}{5} \end{gathered}$$

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 $\tan \left(\angle A\right)=\frac{12}{5}$.