Question

sketch an angle θ in standard position such that θ has the least possible positive measure and the point (-3,4) is on the terminal side. then find the exact values of the six trigonometric functions for θ

u h. the function is undefined.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. tan θ = -4/3 (simplify your answer. type an integer or a fraction.)
b. the function is undefined.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. csc θ = 5/4 (simplify your answer. type an integer or a fraction.)
b. the function is undefined.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. sec θ = (simplify your answer. type an integer or a fraction.)
b. the function is undefined.

Explanation:

Step 1: Recall the definition of secant

The secant of an angle \(\theta\) in standard position, where the point \((x,y)\) is on the terminal side, is defined as \(\sec\theta=\frac{r}{x}\), where \(r = \sqrt{x^{2}+y^{2}}\) (the distance from the origin to the point \((x,y)\)).

Step 2: Identify \(x\), \(y\) and calculate \(r\)

Given the point \((- 3,4)\), we have \(x=-3\) and \(y = 4\).
First, calculate \(r\):
\[r=\sqrt{x^{2}+y^{2}}=\sqrt{(-3)^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25} = 5\]

Step 3: Calculate \(\sec\theta\)

Using the formula \(\sec\theta=\frac{r}{x}\), substitute \(r = 5\) and \(x=-3\):
\[\sec\theta=\frac{5}{-3}=-\frac{5}{3}\]

Answer:

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