Question

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

Short answer

The required trigonometric functions are:

$$\begin{gathered} \sin \theta =\frac{b}{c}=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}csc\theta =\frac{c}{b}=\frac{4}{2\sqrt{3}}=\frac{2}{\sqrt{3}} \\ \cos \theta =\frac{a}{c}=\frac{2}{4}=\frac{1}{2}sec\theta =\frac{c}{a}=\frac{4}{2}=2 \\ \tan \theta =\frac{b}{a}=\frac{2\sqrt{3}}{2}=\sqrt{3}cot\theta =\frac{a}{b}=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}} \end{gathered}$$

Step-by-step solution

Step 1. Given Information

From the figure, we have,$hypotenuse,c=4$and localid="1647160803805" $adjacent.a=2$

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 the adjacent first,

So using Pythagorean Theorem,

$$\begin{gathered} c^2=a^2+b^2 \\ {b}^2=c^2-a^2 \end{gathered}$$

$$\begin{gathered} b^2=4^2-2^2 \\ {b}^2=16-4 \\ {b}^2=12 \end{gathered}$$

$b=2\sqrt{3}$

Now, trigonometric functions are:

$$\begin{gathered} \sin \theta =\frac{b}{c}=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}csc\theta =\frac{c}{b}=\frac{4}{2\sqrt{3}}=\frac{2}{\sqrt{3}} \\ \cos \theta =\frac{a}{c}=\frac{2}{4}=\frac{1}{2}sec\theta =\frac{c}{a}=\frac{4}{2}=2 \\ \tan \theta =\frac{b}{a}=\frac{2\sqrt{3}}{2}=\sqrt{3}cot\theta =\frac{a}{b}=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}} \end{gathered}$$