Question

Given that π < 𝑥 < 2π and tan 𝑥 = 3/4, determine the exact value of cos(2𝑥).

Answer 1

First APPROXIMATE
𝑥 must be in quadrant three, lower left, because cos and sin have the same sign to get the positive tangent.
so 𝑥 is about 180 + tan^-1 3/4 = 180 + 36.86 deg = about 216.86 deg
so 2 𝑥 = about 433.7 deg
433.7 - 360 = 73.7 degrees, in quadrant one so approximately 0.28
now EXACT: (note 3,4,5 triangle)
cos 2𝑥 = cos^2 𝑥 - sin^2 x = (-4/5)^2 - (-3/5)^2
= 16/25 - 9/25 = 7/25 YES, about 0.28

Answer 2

Cos 2x = 4/5 and 90° < x < 180° find six trignometric function?