Question

sketch an angle θ in standard position such that θ has the least possible positive measure and the point (-6,8) is on the terminal side of θ. then find the exact values of the six trigonometric functions for θ. choose the correct graph below. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. sin θ = (simplify your answer. type an integer or a fraction.) b. the function is undefined. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. cos θ = (simplify your answer. type an integer or a fraction.) b. the function is undefined. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. tan θ = (simplify your answer. type an integer or a fraction.) b. the function is undefined.

Explanation:

Step1: Calculate the radius $r$

Given the point $(x,y)=(-6,8)$, use the formula $r = \sqrt{x^{2}+y^{2}}$. So $r=\sqrt{(-6)^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$.

Step2: Find $\sin\theta$

By the definition $\sin\theta=\frac{y}{r}$, substituting $y = 8$ and $r=10$, we get $\sin\theta=\frac{8}{10}=\frac{4}{5}$.

Step3: Find $\cos\theta$

Using the definition $\cos\theta=\frac{x}{r}$, substituting $x=-6$ and $r = 10$, we have $\cos\theta=\frac{-6}{10}=-\frac{3}{5}$.

Step4: Find $\tan\theta$

According to the definition $\tan\theta=\frac{y}{x}$, substituting $x=-6$ and $y = 8$, we obtain $\tan\theta=\frac{8}{-6}=-\frac{4}{3}$.

Step5: Find $\csc\theta$

Since $\csc\theta=\frac{1}{\sin\theta}$, and $\sin\theta=\frac{4}{5}$, then $\csc\theta=\frac{5}{4}$.

Step6: Find $\sec\theta$

As $\sec\theta=\frac{1}{\cos\theta}$, and $\cos\theta=-\frac{3}{5}$, so $\sec\theta=-\frac{5}{3}$.

Step7: Find $\cot\theta$

Because $\cot\theta=\frac{1}{\tan\theta}$, and $\tan\theta=-\frac{4}{3}$, then $\cot\theta=-\frac{3}{4}$.

For the graph, the point $(-6,8)$ is in the second - quadrant. The correct graph is the one where the terminal side of the angle in standard position passes through the point $(-6,8)$ in the second - quadrant.

Answer:

Choose the correct graph: B
A. $\sin\theta=\frac{4}{5}$
A. $\cos\theta=-\frac{3}{5}$
A. $\tan\theta=-\frac{4}{3}$
A. $\csc\theta=\frac{5}{4}$
A. $\sec\theta=-\frac{5}{3}$
A. $\cot\theta=-\frac{3}{4}$