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

$\begin{array}{l}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{40}}{\mathbf{41}}\\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{=}\frac{\mathbf{9}}{\mathbf{41}}\\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{40}}{\mathbf{9}}\\ \mathbf{c}\mathbf{s}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{41}}{\mathbf{40}}\\ \mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\mathbf{41}}{\mathbf{9}}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\theta }\mathbf{=}\frac{\mathbf{9}}{\mathbf{40}}\end{array}$

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 pythagorean theorem,

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

$\begin{array}{l}\text{opp}=40\\ \therefore ad{j}^2={41}^2-{40}^2\\ \therefore ad{j}^2=81\\ \therefore adj=9\end{array}$

Step 3.

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

$\therefore \sin \theta =\frac{40}{41}$

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

$\therefore \cos \theta =\frac{9}{41}$

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

$\therefore \tan \theta =\frac{40}{9}$

The cosecant angle is the reciprocal of sine

$\therefore \csc \theta =\frac{41}{40}$

The secant angle is the reciprocal of cosine

$\therefore \sec \theta =\frac{41}{9}$

The cot angle is the reciprocal of tangent

$\therefore \cot \theta =\frac{9}{40}$