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. (csc \theta = \frac{sqrt{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 denominators.)
b. the function is undefined
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (sec \theta = -sqrt{5})
(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
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (cot \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 denominators.)
b. the function is undefined

Explanation:

Step1: Find a point on the terminal side

Given the equation \(2x + y = 0\) with \(x>0\). Let's choose \(x = 1\), then substitute into the equation: \(2(1)+y = 0\), so \(y=- 2\). So a point on the terminal side is \((x,y)=(1, - 2)\).

Step2: Calculate the radius \(r\)

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

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

The definition of \(\cot\theta\) is \(\cot\theta=\frac{x}{y}\) (for \(y
eq0\)). Here \(x = 1\) and \(y=-2\), so \(\cot\theta=\frac{1}{-2}=-\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\)