Question

Find the exact values of the six trigonometric ratios of the angle$\theta$ in the triangle.

Short answer

The six trigonometric ratios of an angle in the triangle are $\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\frac{\mathbf{4}}{\mathbf{5}}$

$\begin{array}{l}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{=}\frac{\mathbf{3}}{\mathbf{5}}\\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\\ \mathbf{c}\mathbf{s}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{5}}{\mathbf{4}}\\ \mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{5}}{\mathbf{3}}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\theta }\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}\end{array}$

Step-by-step solution

Step 1. Finding values for trigonometric functions.

From the given right triangle,

$\begin{array}{l}\text{hyp}=5\\ adj=3\\ opp=4\end{array}$

The sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.

$\therefore \sin \theta =\frac{4}{5}$

The cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.

$\therefore \cos \theta =\frac{3}{5}$

The tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.

$\therefore \tan \theta =\frac{4}{3}$

The cosecant angle is the reciprocal of sine

$\therefore \csc \theta =\frac{5}{4}$

The secant angle is the reciprocal of cosine

$\therefore \sec \theta =\frac{5}{3}$

The cot angle is the reciprocal of tangent

$\therefore \cot \theta =\frac{3}{4}$