Question

the point given below is on the terminal side of an angle \\( \theta \\) in standard position. find the exact value of each of the six trigonometric functions of \\( \theta \\)
\\\

$$\begin{array}{l}(12,-9)\\\\\\sin \\theta=\\square\\end{array}$$

\\
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression )

Explanation:

Step1: Find the value of \(r\)

For a point \((x,y)\) on the terminal side of an angle \(\theta\), \(r=\sqrt{x^{2}+y^{2}}\). Here \(x = 12\) and \(y=-9\), so \(r=\sqrt{12^{2}+(-9)^{2}}=\sqrt{144 + 81}=\sqrt{225}=15\).

Step2: Calculate \(\sin\theta\)

The formula for \(\sin\theta=\frac{y}{r}\). Substituting \(y=-9\) and \(r = 15\), we get \(\sin\theta=\frac{-9}{15}=-\frac{3}{5}\).

Answer:

\(-\frac{3}{5}\)