Question

find the exact value of each of the six trigonometric functions of \\( \theta \\), if \\( (-8,-7) \\) is a point on the terminal side of angle \\( \theta \\).
\\\sin \theta=\frac{-7 \sqrt{113}}{113}\\(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\\\cos \theta=\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, \( x = - 8\) and \( y=-7 \), so \( r=\sqrt{(-8)^{2}+(-7)^{2}}=\sqrt{64 + 49}=\sqrt{113} \).

Step2: Calculate \( \cos\theta \)

The formula for \( \cos\theta \) is \( \cos\theta=\frac{x}{r} \). Substituting \( x=-8 \) and \( r = \sqrt{113} \), we get \( \cos\theta=\frac{-8}{\sqrt{113}} \). Rationalizing the denominator (multiplying numerator and denominator by \( \sqrt{113} \)), we have \( \cos\theta=\frac{-8\sqrt{113}}{113} \).

Answer:

\(\frac{-8\sqrt{113}}{113}\)