Question

Let theta be an angle in quadrant 2 such that sec theta = -5/4. Find the exact values of cot theta and sin theta.

Options:
A. cot theta = -4/5, sin theta = 3/5
B. cot theta = 5/4, sin theta = -3/5
C. cot theta = 3/5, sin theta = -4/5
D. cot theta = -3/5, sin theta = 4/5

Answer 1

Given that sec theta = -5/4 in quadrant 2, we find sin theta to be 3/5 and cot theta to be -4/3 using the Pythagorean identity and the definition of cotangent.

Step-by-step explanation:

The given information states that theta is an angle in quadrant 2 where sec theta = -5/4. Since sec theta is the reciprocal of cos theta, we have cos theta = -4/5. In the second quadrant, sin theta is positive, and cot theta is the reciprocal of tan theta which is sin theta over cos theta.

First, we find sin theta using the Pythagorean identity sin²(theta) + cos²(theta) = 1. Since cos theta is -4/5, we calculate sin theta to be positive as we are in the second quadrant. So, sin theta = √(1 - cos²(theta)) = √(1 - (-4/5)²) = √(1 - 16/25) = √(9/25) = 3/5.

Next, to find cot theta, we use the definition cot theta = cos theta / sin theta. Therefore, cot theta = (-4/5) / (3/5) = -4/3.

Thus, the correct answer is A. cot theta = -4/3, sin theta = 3/5.