Question

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

Explanation:

Step1: Find the opposite side

Let the opposite side be $y$. Using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 85$ and $a=84$. So $y=\sqrt{85^{2}-84^{2}}=\sqrt{(85 + 84)(85 - 84)}=\sqrt{169}=13$.

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

$\csc\theta=\frac{1}{\sin\theta}=\frac{85}{13}$.

Step6: Calculate secant

$\sec\theta=\frac{1}{\cos\theta}=\frac{85}{84}$.

Step7: Calculate cotangent

$\cot\theta=\frac{1}{\tan\theta}=\frac{84}{13}$.

Answer:

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