Question

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

b. the function is undefined

select the correct choice below and, if necessary, fill in the answer box to complete your choice

a.
$cos \theta = -\frac{\sqrt{5}}{5}$
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize a

b. the function is undefined

select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. $\tan \theta = -2$
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize al

b. the function is undefined

select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. $csc \theta = \square$
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize all

Explanation:

Step1: Find a point on the terminal side

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

Step2: Calculate the radius \(r\)

For a point \((x,y)\) in the plane, the radius \(r\) (distance from the origin) is given by the formula \(r=\sqrt{x^{2}+y^{2}}\).
Substitute \(x=-1\) and \(y = 2\) into the formula:
\(r=\sqrt{(-1)^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\)

Step3: Recall the definition of \(\csc\theta\)

The cosecant function is defined as \(\csc\theta=\frac{r}{y}\) (where \(y
eq0\)).

Step4: Substitute the values of \(r\) and \(y\)

We know that \(r = \sqrt{5}\) and \(y=2\). So:
\(\csc\theta=\frac{\sqrt{5}}{2}\)

Answer:

\(\frac{\sqrt{5}}{2}\)