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.
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$sin\theta=$
$cos\theta=$
$\tan\theta=$
$csc\theta=$
$sec\theta=$
$cot\theta=$

Explanation:

Step1: Find the opposite side

Let the hypotenuse $c = 8\sqrt{7}$ and the adjacent side $a=6\sqrt{7}$. Using the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, we can find the opposite side $b$.
\[b=\sqrt{c^{2}-a^{2}}=\sqrt{(8\sqrt{7})^{2}-(6\sqrt{7})^{2}}=\sqrt{448 - 252}=\sqrt{196}=14\]

Step2: Calculate $\sin\theta$

By the definition of sine, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So $\sin\theta=\frac{14}{8\sqrt{7}}=\frac{\sqrt{7}}{4}$.

Step3: Calculate $\cos\theta$

By the definition of cosine, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So $\cos\theta=\frac{6\sqrt{7}}{8\sqrt{7}}=\frac{3}{4}$.

Step4: Calculate $\tan\theta$

By the definition of tangent, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So $\tan\theta=\frac{14}{6\sqrt{7}}=\frac{\sqrt{7}}{3}$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{1}{\sin\theta}$, $\csc\theta=\frac{4}{\sqrt{7}}=\frac{4\sqrt{7}}{7}$.

Step6: Calculate $\sec\theta$

Since $\sec\theta=\frac{1}{\cos\theta}$, $\sec\theta=\frac{4}{3}$.

Step7: Calculate $\cot\theta$

Since $\cot\theta=\frac{1}{\tan\theta}$, $\cot\theta=\frac{3}{\sqrt{7}}=\frac{3\sqrt{7}}{7}$.

Answer:

$\sin\theta=\frac{\sqrt{7}}{4}$, $\cos\theta=\frac{3}{4}$, $\tan\theta=\frac{\sqrt{7}}{3}$, $\csc\theta=\frac{4\sqrt{7}}{7}$, $\sec\theta=\frac{4}{3}$, $\cot\theta=\frac{3\sqrt{7}}{7}$