Question

the equation, with a restriction on x, is the terminal side of an angle θ in standard position.
2x + y = 0, x ≤ 0

give the exact values of the six trigonometric functions of θ.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○ a. sin θ =
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize all denominators
○ b. the function is undefined.

Explanation:

Step1: Find a point on the terminal side

Given the equation \(2x + y = 0\) with \(x\leq0\). Let's choose a value for \(x\) (since \(x\leq0\), we can pick \(x=- 1\)). Substitute \(x = - 1\) into the equation:
\(2(-1)+y = 0\)
\(-2 + y=0\), so \(y = 2\). So the point \((-1,2)\) lies on the terminal side of the angle \(\theta\).

Step2: Calculate the radius \(r\)

The formula for \(r\) (the distance from the origin to the point \((x,y)\)) is \(r=\sqrt{x^{2}+y^{2}}\). For the point \((-1,2)\), we have \(x=-1\) and \(y = 2\).
\(r=\sqrt{(-1)^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\)

Step3: Calculate \(\sin\theta\)

The formula for \(\sin\theta\) is \(\sin\theta=\frac{y}{r}\). We know that \(y = 2\) and \(r=\sqrt{5}\).
\(\sin\theta=\frac{2}{\sqrt{5}}\)
Rationalize the denominator: \(\frac{2}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{2\sqrt{5}}{5}\)

Answer:

A. \(\sin\theta=\frac{2\sqrt{5}}{5}\)