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 = -3/5 (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) cosθ = 4/5 (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) tanθ = -3/4 (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) cotθ = -4/3 (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) secθ =

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 \(\sec\theta\)

The formula for \(\sec\theta=\frac{r}{x}\) (since \(\sec\theta=\frac{1}{\cos\theta}\) and \(\cos\theta=\frac{x}{r}\)).
Substitute \(r = 15\) and \(x = 12\) into the formula: \(\sec\theta=\frac{15}{12}=\frac{5}{4}\)

Answer:

\(\frac{5}{4}\)