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

$\begin{array}{l}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{3}}{\sqrt{\mathbf{13}}}\\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{=}\frac{\mathbf{2}}{\sqrt{\mathbf{13}}}\\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}\\ \mathbf{c}\mathbf{s}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\sqrt{\mathbf{13}}}{\mathbf{3}}\\ \mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\mathbf{=}\frac{\sqrt{\mathbf{13}}}{\mathbf{2}}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\theta }\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\end{array}$

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,

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

$\begin{array}{l}\text{adj}=2\\ \text{opp}=3\\ \therefore {\text{hyp}}^2=3^2+2^2\\ \therefore {\text{hyp}}^2=13\\ \therefore \text{hyp}=\sqrt{13}\end{array}$

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.

$\sin \theta =\frac{3}{\sqrt{13}}$

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

$\cos \theta =\frac{2}{\sqrt{13}}$

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

$\tan \theta =\frac{3}{2}$

The cosecant angle is the reciprocal of sine

$\csc \theta =\frac{\sqrt{13}}{3}$

The secant angle is the reciprocal of cosine

$\sec \theta =\frac{\sqrt{13}}{2}$

The cot angle is the reciprocal of tangent

$\cot \theta =\frac{2}{3}$