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 adjacent side $a = 8$, the hypotenuse $c=8\sqrt{2}$. Using the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, we have $(8\sqrt{2})^{2}=8^{2}+b^{2}$. Then $128 = 64 + b^{2}$, and $b^{2}=128 - 64=64$, so $b = 8$.

Step2: Calculate sine

The sine function is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So $\sin\theta=\frac{8}{8\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Step3: Calculate cosine

The cosine function is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So $\cos\theta=\frac{8}{8\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Step4: Calculate tangent

The tangent function is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So $\tan\theta=\frac{8}{8}=1$.

Step5: Calculate cosecant

The cosecant function is the reciprocal of sine. So $\csc\theta=\frac{1}{\sin\theta}=\sqrt{2}$.

Step6: Calculate secant

The secant function is the reciprocal of cosine. So $\sec\theta=\frac{1}{\cos\theta}=\sqrt{2}$.

Step7: Calculate cotangent

The cotangent function is the reciprocal of tangent. So $\cot\theta=\frac{1}{\tan\theta}=1$.

Answer:

$\sin\theta=\frac{\sqrt{2}}{2}$, $\cos\theta=\frac{\sqrt{2}}{2}$, $\tan\theta = 1$, $\csc\theta=\sqrt{2}$, $\sec\theta=\sqrt{2}$, $\cot\theta = 1$