Question

sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. then find the values of the six trigonometric functions for the angle. rationalize denominators when applicable. (-2√5,4)
choose the correct sketch of the angle below
select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. sin θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function is undefined
select the correct choice below and, if necessary, fill in the answer box to complete your choice

Explanation:

Step1: Calculate the radius $r$

Given the point $(x,y)=(- 2\sqrt{5},4)$, use the formula $r=\sqrt{x^{2}+y^{2}}$. So $r = \sqrt{(-2\sqrt{5})^{2}+4^{2}}=\sqrt{20 + 16}=\sqrt{36}=6$.

Step2: Find $\sin\theta$

The formula for $\sin\theta$ is $\sin\theta=\frac{y}{r}$. Substitute $y = 4$ and $r=6$ into the formula, we get $\sin\theta=\frac{4}{6}=\frac{2}{3}$.

For the sketch of the angle: The point $(-2\sqrt{5},4)$ has a negative $x$-coordinate and a positive $y$-coordinate, so the angle $\theta$ in standard - position with this point on its terminal side lies in the second quadrant.

Answer:

For the sketch: The angle in the second - quadrant is the correct one (you need to visually identify the sketch with the terminal side in the second quadrant among the options).
For $\sin\theta$: A. $\sin\theta=\frac{2}{3}$