Question

Given theta is an angle in Quadrant III such that sin(theta) =-3/5 How does one find the exact values of sec(theta) and cot(theta)?

Answer 1

`sec theta = -5/4; " " cot theta = 4/3`

Explanation:

Given: `theta` is an angle in Quadrant III. `sin theta = -3/5`

In Quadrant III, cosine (the `x`-value of the unit circle) and sine (the `y`-value of the unit circle) are both negative and tangent is positive.

`sin theta = -3/5 = ("opposite side")/("hypotenuse")`

To find the adjacent side of the triangle use the Pythagorean Theorem: `x^2 + (-3)^2 = 5^2`

`x = +-sqrt(5^2 - (-3)^2) = +- sqrt(25 - 9) = +-sqrt(16) = +-4`

Since we are in the third quadrant `x = -4`

One way to find the secant and cotangent is to use the inverse identities:

`sec theta = 1/(cos theta) = 1/(-4/5) = 1 * 5/-4 = -5/4`

`cot theta = 1/(tan theta) = 1/(-3/-4) = 1 * (-4)/-3 = 4/3`