Question

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

Short answer

The solution is $\sin \theta =\frac{3}{5},\cos \theta =\frac{4}{5},\tan \theta =\frac{3}{4},csc\theta =\frac{5}{3},sec\theta =\frac{5}{4},cot\theta =\frac{4}{3}$

Step-by-step solution

Step 1. Given information 

Adjacent$=4$and opposite$=3$

Step 2. Calculation 

Let $a=4$, $b=3$and role="math" localid="1647102231651" $c=$hypotenuse.

Now, role="math" localid="1647102404943" $c^2=a^2+b^2$ Pythagorean Theorem

$\sqrt{c^2}=\sqrt{4^2+3^2}$

$c=\sqrt{16+9}$

$c=\sqrt{25}$

$c=5$

Step 3. Calculation 

The value of each of the six trigonometric functions are -
$\sin \theta =\frac{b}{c}=\frac{3}{5}$ $\cos \theta =\frac{a}{c}=\frac{4}{5}$ $\tan \theta =\frac{b}{a}=\frac{3}{4}$

$csc\theta =\frac{c}{b}=\frac{5}{3}$ $sec\theta =\frac{c}{a}=\frac{5}{4}$ $cot\theta =\frac{a}{b}=\frac{4}{3}$