tan 𝑥 = −5 and sin 𝑥 > 0, find sin 2x

Answer 1

Since tan(x) = -5 and sin(x) > 0 we must have an angle in quadrant 2 (sin(x) is positive there but tan(x) is negative there). Since tan(x) = opposite/adjacent and tan(x) = - 5/1 then the opposite side must be 5 and the adjacent side must be -1 like this:

Using the Pythagorean Theorem, we can solve for h: h2 = 5² + (-1)² or h = √26.

The double angle identity for sine is sin(2x) = 2sin(x)cos(x)

In this case sin(x) = 5/√26 and cos(x) = -1/√26 so sin(2x) = 2(5/√26)(-1/√26) = -10/26 = -5/13