Mustafa S. answered • 07/24/19

High School AP Math/Science Teacher with SAT prep experience

Hello Alyssa,

I am going to assume the question says cos(x) = -4/5, but correct me if I am wrong.

If cos(x) = -4/5 it means there is a right triangle with adjacent side of 4, and hypotenuse of 5. The negative is because this triangle is located in the third quadrant (180^{o} < x < 270^{o}), in which cos is negative. The remaining leg of the triangle (the opposite side) has to be 3 (from the Pythagorean theorem). So we have a 3, 4, 5 right triangle. Tangent is opposite/adjacent, which in this case is 3/4. We also know that this number should be positive because tangent is positive in the third quadrant.

To find tan(2x) we need to use the trig identity:

tan(2x) = 2tan(x) / (1-tan^{2}(x))

Substituting our value of tan(x) = 3/4 into this identity, we get:

tan(2x) = 2(3/4) / (1 - (3/4)^{2})

tan(2x) = (3/2) / (1 - 9/16)

tan(2x) = (3/2) / (7/16)

tan(2x) = (3/2) * (16/7)

tan(2x) = 48/14 = 24/7