Question

the point given below is on the terminal side of an angle θ in standard position. find the exact value of each of the six trigonometric functions of θ (12, - 9) sinθ = - \frac{3}{5} (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression ) cosθ = \frac{4}{5} (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression ) tanθ = - \frac{3}{4} (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression ) cotθ = - \frac{4}{3} (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression ) secθ = \frac{5}{4} (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) cscθ =

Explanation:

Step1: Recall the formula for \(r\)

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

Step2: Recall the formula for \(\csc\theta\)

The formula for \(\csc\theta=\frac{r}{y}\). Since \(r = 15\) and \(y=-9\), then \(\csc\theta=\frac{15}{-9}=-\frac{5}{3}\).

Answer:

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