Question

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

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a.
sin θ = (\frac{2sqrt{5}}{5})
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. ration
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 = (-\frac{sqrt{5}}{5})
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. ration
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 = (square)
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. ration
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 \((-1,2)\) is 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 \((-1,2)\):
\(r=\sqrt{(-1)^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\)

Step3: Calculate \(\tan\theta\)

The formula for \(\tan\theta\) is \(\tan\theta=\frac{y}{x}\) (where \(x
eq0\)). For the point \((-1,2)\), \(x=-1\) and \(y = 2\):
\(\tan\theta=\frac{2}{-1}=- 2\)

Answer:

\(\tan\theta=-2\) (So the correct choice is A with \(\tan\theta=-2\))