Question

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

Short answer

The solution is $\sin \theta =\frac{5}{13},\cos \theta =\frac{12}{13},\tan \theta =\frac{5}{12},csc\theta =\frac{13}{5},sec\theta =\frac{13}{12},cot\theta =\frac{12}{5}$

Step-by-step solution

Step 1. Given information 

Adjacent$=12$and opposite$=5$

Step 2. Calculation 

Let $a=12$, $b=5$and $c=$hypotenuse.

Now, role="math" localid="1647102386605" $c^2=a^2+b^2$ Pythagorean Theorem

role="math" localid="1647102395725" $\sqrt{c^2}=\sqrt{{12}^2+5^2}$

$c=\sqrt{144+25}$

$c=\sqrt{169}$

$c=13$

Step 3. Calculation 

The value of each of the six trigonometric functions are -

$\sin \theta =\frac{b}{c}=\frac{5}{13}$ $\cos \theta =\frac{a}{c}=\frac{12}{13}$ $\tan \theta =\frac{b}{a}=\frac{5}{12}$ $csc\theta =\frac{c}{b}=\frac{13}{5}$ $sec\theta =\frac{c}{a}=\frac{13}{12}$ $cot\theta =\frac{a}{b}=\frac{12}{5}$