Question

If cos x = - and x lies in the third quadrant, find the values of the other five trigonometric functions.

A. sin x = -, tan x = -, cot x = -, sec x = -, csc x = -
B. sin x = -, tan x = -, cot x = -, sec x = -, csc x = -
C. sin x = -, tan x = -, cot x = -, sec x = -, csc x = -
D. sin x = -, tan x = -, cot x = -, sec x = -, csc x = -

Answer 1

In the third quadrant, the values of the trigonometric functions are sin x = -√3/2, tan x = √3/2, cot x = 2/√3, sec x = 1, and csc x = -2/√3.

Step-by-step explanation:

In the third quadrant, the cosine function is negative, which means that cos x = -.

To find the values of the other five trigonometric functions, we can use the unit circle. In the third quadrant, the sine function is positive, so sin x = -√3/2. The tangent function is sin x over cos x, so tan x = (-√3/2) / (-1) = √3/2.

The cotangent function is the reciprocal of the tangent, so cot x = 1 / tan x = 2/√3. The secant function is the reciprocal of the cosine, so sec x = 1 / cos x = -1 / -1 = 1. Lastly, the cosecant function is the reciprocal of the sine, so csc x = 1 / sin x = -2/√3.