Question

use the information contained in the figure to determine the values of the six trigonometric functions of θ. write the exact answers and simplify. do not round.

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 13$ and $b = 84$. So $c=\sqrt{13^{2}+84^{2}}=\sqrt{169 + 7056}=\sqrt{7225}=85$.

Step2: Calculate sine

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{13}{85}$.

Step3: Calculate cosine

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{84}{85}$.

Step4: Calculate tangent

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{13}{84}$.

Step5: Calculate cosecant

$\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{85}{13}$.

Step6: Calculate secant

$\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{85}{84}$.

Step7: Calculate cotangent

$\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{84}{13}$.

Answer:

$\sin\theta=\frac{13}{85}$, $\cos\theta=\frac{84}{85}$, $\tan\theta=\frac{13}{84}$, $\csc\theta=\frac{85}{13}$, $\sec\theta=\frac{85}{84}$, $\cot\theta=\frac{84}{13}$