Question

If cos(θ) = -5/7 and θ is in quadrant III, find tan(θ), cot(θ), csc(θ).

a) tan(θ) = -5/12, cot(θ) = -12/5, csc(θ) = -7/5
b) tan(θ) = -7/5, cot(θ) = -5/12, csc(θ) = -12/5
c) tan(θ) = -5/7, cot(θ) = -7/5, csc(θ) = -5/7
d) tan(θ) = -7/5, cot(θ) = -7/5, csc(θ) = -5/7

Answer 1

To find the values of tan(θ), cot(θ), and csc(θ) when cos(θ) = -5/7 and θ is in quadrant III, you can use the formulas tan(θ) = sin(θ) / cos(θ), cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ), and csc(θ) = 1 / sin(θ).

Step-by-step explanation:

To find tan(θ), we can use the formula tan(θ) = sin(θ) / cos(θ). Since we know that cos(θ) = -5/7, we can substitute that value.

First, we need to find sin(θ). Since θ is in quadrant III, which is the bottom left quadrant, we know that sin(θ) is negative. We can use the formula sin(θ) = -sqrt(1 - cos^2(θ)) to find sin(θ).

Next, we can substitute the values into the formulas to find tan(θ), cot(θ), and csc(θ):

tan(θ) = sin(θ) / cos(θ)

cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)

csc(θ) = 1 / sin(θ)