Question

If cos(x)=-1/4 and tan(x)<0 what is sin(2x)​

Answer 1

By definition of tangent,

tan(x) = sin(x) / cos(x)

so if tan(x) < 0, and we're given cos(x) = -1/4 < 0, then it follows that sin(x) > 0.

Recall the Pythagorean identity:

cos²(x) + sin²(x) = 1   →   sin(x) = + √(1 - cos²(x))

Then

sin(x) = √(1 - (-1/4)²) = √(15/16) = √(15)/4

Recall the double angle identity:

sin(2x) = 2 sin(x) cos(x)

Then

sin(2x) = 2 • √(15)/4 • (-1/4) = -2√(15)/16 = -√(15)/8