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

$\begin{array}{l}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{15}}{\mathbf{17}}\\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{=}\frac{\mathbf{8}}{\mathbf{17}}\\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{15}}{\mathbf{8}}\\ \mathbf{c}\mathbf{s}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{17}}{\mathbf{15}}\\ \mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{17}}{\mathbf{8}}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\theta }\mathbf{=}\frac{\mathbf{8}}{\mathbf{15}}\end{array}$

Step-by-step solution

Step 1. Given information

Given: A right angle triangle with$\theta$ as its acute angle.
To find the six trigonometric ratios.
Identifying the sides of the right triangle.
And finding the required ratios.

Step 2. Finding adjacent side.

By using Pythagoreantheorem,

$hy{p}^2=op{p}^2+ad{j}^2$

$\begin{array}{l}\begin{array}{l}\text{adj}=8\\ \text{opp}=15\end{array}\\ \therefore hy{p}^2={15}^2+8^2\\ \therefore hy{p}^2=289\\ \therefore hyp=17\end{array}$

Step 3. Finding trigonometric function values.

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

$\therefore \sin \theta =\frac{15}{17}$

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

$\therefore \cos \theta =\frac{8}{17}$

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

$\therefore \tan \theta =\frac{15}{8}$

The cosecant angle is the reciprocal of sine

$\therefore \csc \theta =\frac{17}{15}$

The secant angle is the reciprocal of cosine

$\therefore \sec \theta =\frac{17}{8}$

The cot angle is the reciprocal of tangent

$\therefore \cot \theta =\frac{8}{15}$