Question

use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ.

Explanation:

Step1: Apply Pythagorean Theorem

The Pythagorean Theorem is $a^{2}+b^{2}=c^{2}$. Given $a = 9$ and $b = 12$, we substitute these values: $c=\sqrt{a^{2}+b^{2}}=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$.

Step2: Define sine function

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}=\frac{9}{15}=\frac{3}{5}$.

Step3: Define cosine function

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}=\frac{12}{15}=\frac{4}{5}$.

Step4: Define tangent function

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{9}{12}=\frac{3}{4}$.

Step5: Define cosecant function

$\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{c}{a}=\frac{15}{9}=\frac{5}{3}$.

Step6: Define secant function

$\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{c}{b}=\frac{15}{12}=\frac{5}{4}$.

Step7: Define cotangent function

$\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{b}{a}=\frac{12}{9}=\frac{4}{3}$.

Answer:

The length of the missing side $c = 15$.
$\sin\theta=\frac{3}{5}$, $\cos\theta=\frac{4}{5}$, $\tan\theta=\frac{3}{4}$, $\csc\theta=\frac{5}{3}$, $\sec\theta=\frac{5}{4}$, $\cot\theta=\frac{4}{3}$