Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator.
240°

b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. (csc 240^{circ}=)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. (sec 240^{circ}=)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. (cot 240^{circ}=)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.

Explanation:

Step1: Find the reference angle

The reference angle for \(240^{\circ}\) is \(240^{\circ}- 180^{\circ}=60^{\circ}\).

Step2: Determine the sign of trigonometric functions

Since \(240^{\circ}\) is in the third quadrant, \(\sin\theta<0\), \(\cos\theta<0\), \(\tan\theta>0\).

• For \(\csc240^{\circ}\):

We know that \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin240^{\circ}=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}\), so \(\csc240^{\circ}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\).

• For \(\sec240^{\circ}\):

We know that \(\sec\theta=\frac{1}{\cos\theta}\), and \(\cos240^{\circ}=-\cos60^{\circ}=-\frac{1}{2}\), so \(\sec240^{\circ}=\frac{1}{-\frac{1}{2}}=- 2\).

• For \(\cot240^{\circ}\):

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), \(\cos240^{\circ}=-\frac{1}{2}\), \(\sin240^{\circ}=-\frac{\sqrt{3}}{2}\), so \(\cot240^{\circ}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{\sqrt{3}}{3}\).

Answer:

A. \(\csc240^{\circ}=-\frac{2\sqrt{3}}{3}\)
A. \(\sec240^{\circ}=-2\)
A. \(\cot240^{\circ}=\frac{\sqrt{3}}{3}\)