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. tan θ = -2
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize all den
b. the function is undefined
select the correct choice below and, if necessary, fill in the answer box to complete your choice

a.
csc θ = √5 / 2
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize all de
b. the function is undefined.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. sec θ =
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed. rationalize all d

Explanation:

Step1: Find a point on the terminal side

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

Step2: Calculate \(r\) (distance from origin)

The formula for \(r\) is \(r=\sqrt{x^{2}+y^{2}}\). Substitute \(x=-1\) and \(y = 2\):
\(r=\sqrt{(-1)^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\)

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

The secant function is defined as \(\sec\theta=\frac{r}{x}\) (where \(x
eq0\)).

Step4: Substitute \(r\) and \(x\) values

We have \(r = \sqrt{5}\) and \(x=-1\). So,
\(\sec\theta=\frac{\sqrt{5}}{-1}=-\sqrt{5}\)

Answer:

\(-\sqrt{5}\)