Question

Find the exact value of the six trigonometric functions of the angle $\theta$in each figure.

Short answer

The required six trigonometric functions are:

$$\begin{gathered} \sin \theta =\frac{b}{c}=\frac{3}{4}csc\theta =\frac{c}{b}=\frac{4}{3} \\ \cos \theta =\frac{a}{c}=\frac{\sqrt{7}}{4}sec\theta =\frac{c}{a}=\frac{4}{\sqrt{7}} \\ \tan \theta =\frac{b}{a}=\frac{3}{\sqrt{7}}cot\theta =\frac{a}{b}=\frac{\sqrt{7}}{3} \end{gathered}$$

Step-by-step solution

Step 1. Given Information

From the figure, we have,

hypotenuse,$c=4$and opposite,$b=3$.

Step 2. Formula Used

Pythagorean Theorem:

$c^2=a^2+b^2$, where a, b and c are adjacent, opposite and hypotenuse respectively.

Trigonometric Functions:

$$\begin{gathered} \sin \theta =\frac{b}{c}csc\theta =\frac{c}{b} \\ \cos \theta =\frac{a}{c}sec\theta =\frac{c}{a} \\ \tan \theta =\frac{b}{a}cot\theta =\frac{a}{b} \end{gathered}$$

Step 3. Calculation

We need to find out the adjacent first.

By Pythagorean Theorem,

$$\begin{gathered} c^2=a^2+b^2 \\ {a}^2=4^2-3^2 \\ {a}^2=16-9 \\ {a}^2=7 \\ a=\sqrt{7} \end{gathered}$$

Now, the six trigonometric functions are:

$$\begin{gathered} \sin \theta =\frac{b}{c}=\frac{3}{4}csc\theta =\frac{c}{b}=\frac{4}{3} \\ \cos \theta =\frac{a}{c}=\frac{\sqrt{7}}{4}sec\theta =\frac{c}{a}=\frac{4}{\sqrt{7}} \\ \tan \theta =\frac{b}{a}=\frac{3}{\sqrt{7}}cot\theta =\frac{a}{b}=\frac{\sqrt{7}}{3} \end{gathered}$$