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 = (\frac{-2sqrt{13}}{13})
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

cos θ = (square)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Find the radius \( r \)

For a point \((x, y)\) on the terminal side of an angle \(\theta\), \( r=\sqrt{x^{2}+y^{2}} \). Here, \( x = 9 \), \( y=-6 \).
\[
r=\sqrt{9^{2}+(-6)^{2}}=\sqrt{81 + 36}=\sqrt{117}=\sqrt{9\times13}=3\sqrt{13}
\]

Step2: Calculate \( \cos\theta \)

The formula for \( \cos\theta \) is \( \cos\theta=\frac{x}{r} \). Substitute \( x = 9 \) and \( r = 3\sqrt{13} \).
\[
\cos\theta=\frac{9}{3\sqrt{13}}=\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}
\]

Answer:

\(\frac{3\sqrt{13}}{13}\)