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 = \frac{2\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 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.
cos 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 \(x\) that satisfies \(x\leq0\). Let's take \(x=-1\) (since \(x\) should be non - positive). Substitute \(x = - 1\) into the equation \(2x + y=0\), we get \(2\times(-1)+y = 0\), which simplifies to \(-2 + y=0\), so \(y = 2\). So the point \((x,y)=(-1,2)\) lies on the terminal side of the angle \(\theta\).

Step2: Calculate the radius \(r\)

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

Step3: Calculate \(\cos\theta\)

The definition of \(\cos\theta\) is \(\cos\theta=\frac{x}{r}\). We know that \(x=-1\) and \(r = \sqrt{5}\), so \(\cos\theta=\frac{-1}{\sqrt{5}}\). Rationalizing the denominator (multiplying the numerator and denominator by \(\sqrt{5}\)), we get \(\cos\theta=-\frac{\sqrt{5}}{5}\).

Answer:

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