Use identities to find the value of the expression

Question

Find sec theta and sin theta if cosine theta = 3/4 and tan theta < 0.

From what i know, this problem takes place in quadrant 4

Answer 1

The tangent is negative in quadrants II and IV. The cosine is positive in quadrants I and IV. So yes, theta is in quadrant IV.

The secant is easy:

sec(θ) = 1/cos(θ) = 1/(3/4) = 4/3

To find sin(θ), use the Pythagorean Identity:

sin²(θ) + cos²(θ) = 1

sin²(θ) = 1 - cos²(θ)

sin²(θ) = 1 - (3/4)²

sin²(θ) = 1 - 9/16 = 7/16

sin(θ) = ±√(7/16) = ±(1/4)√7

Is the sine positive or negative in Q IV?