Question

Trigonometric Ratios 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

$\begin{array}{l}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{7}}{\mathbf{25}}\mathbf{,}\\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{=}\frac{\mathbf{24}}{\mathbf{25}}\mathbf{,}\\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{7}}{\mathbf{24}}\mathbf{,}\\ \mathbf{c}\mathbf{s}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{25}}{\mathbf{7}}\mathbf{,}\\ \mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{25}}{\mathbf{24}}\mathbf{,}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\theta }\mathbf{=}\frac{\mathbf{24}}{\mathbf{7}}\end{array}$

Step-by-step solution

Step 1. Finding the trigonometric values.

From the given right triangle,

$\begin{array}{c}hyp=25\\ adj=24\\ opp=7\end{array}$

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

$\therefore \sin \theta =\frac{7}{25}$

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

$\therefore \cos \theta =\frac{24}{25}$

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

$\therefore \tan \theta =\frac{7}{24}$

The cosecant angle is the reciprocal of sine

$\therefore \csc \theta =\frac{25}{7}$

The secant angle is the reciprocal of cosine

$\therefore \sec \theta =\frac{25}{24}$

The cot angle is the reciprocal of tangent

$\therefore \cot \theta =\frac{24}{7}$