find sin theta, given sec theta= 4/3 and tan theta<0

Question

I know that theta must be in the fourth quadrant but I am lost on what identities to use for solving sin theta.

Answer 1

the secant is the inverse of the cosine

so cos theta = 1/(sec theta) = 3/4

sin ^2 theta + cos^2 theta = 1 (basic trig identity)

sin ^2 theta + 9/16 = 1

sin ^2 theta = 7/16

sin theta = 7^0.5/4

since angle located in Quadrant IV, sin theta = -7^0.5/4

Answer 2

You are correct that it is in Quadrant IV. Consider a right triangle including the origin, the point (3,0), and a point on the terminal ray of theta that is 4 units from the origin. By the Pythagorean theorem, the opposite side must have a length of radical 7. It is below the x-axis, so we use negative root 7. The sin of this same angle is then -(√7)/4