Question

find the exact value of each of the six trigonometric functions of θ, if (9, −6) is a point on the terminal side of angle θ.
sin θ=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Identify x, y, and r

Given the point \((9, -6)\) on the terminal side of \(\theta\), we have \(x = 9\), \(y = -6\). To find \(r\) (the distance from the origin to the point), we use the formula \(r=\sqrt{x^{2}+y^{2}}\).
\[
r=\sqrt{9^{2}+(-6)^{2}}=\sqrt{81 + 36}=\sqrt{117}=\sqrt{9\times13}=3\sqrt{13}
\]

Step2: Calculate \(\sin\theta\)

The formula for \(\sin\theta\) is \(\sin\theta=\frac{y}{r}\). Substituting \(y = -6\) and \(r = 3\sqrt{13}\) into the formula:
\[
\sin\theta=\frac{-6}{3\sqrt{13}}=\frac{-2}{\sqrt{13}}
\]
Rationalizing the denominator (multiplying numerator and denominator by \(\sqrt{13}\)):
\[
\sin\theta=\frac{-2\sqrt{13}}{13}
\]

Answer:

\(\frac{-2\sqrt{13}}{13}\)