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 $\sin \left(\angle A\right)=\frac{4}{5}$,$\cos \left(\angle A\right)=\frac{3}{5}$and $\tan \left(\angle A\right)=\frac{4}{3}$.

Step-by-step solution

Step 1. Given.

The values of the three trigonometric ratios 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 A.

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 4, 3, and 5 respectively.

Therefore,

$$\begin{gathered} \sin \left(\angle A\right)=\frac{\text{legopposite}\angle \text{A}}{\text{hypotenuse}} \\ =\frac{4}{5} \end{gathered}$$

$$\begin{gathered} \cos \left(\angle A\right)=\frac{\text{legadjacentto}\angle \text{A}}{\text{hypotenuse}} \\ =\frac{3}{5} \end{gathered}$$

$$\begin{gathered} \tan \left(\angle A\right)=\frac{\text{legopposite}\angle \text{A}}{\text{legadjacentto}\angle \text{A}} \\ =\frac{4}{3} \end{gathered}$$

Therefore, the values of the three trigonometric ratios for angle A are $\sin \left(\angle A\right)=\frac{4}{5}$,$\cos \left(\angle A\right)=\frac{3}{5}$ and $\tan \left(\angle A\right)=\frac{4}{3}$.