Find the exact value of sin(2x) assuming that sin(x) = -5/13 and that π < x < 3π/2

Answer 1

sin2x = exactly 120/169

5/13 implies a 5-12-13 right triangle

as sinx = opposite side/hypotenuse

13 squared - 5 squared = 169 -25 = 144 = 12 squared. 12 is the adjacent side and

cosx = adjacent side/hypotenuse = 12/13 with a negative sine in the 3rd quadrant = -12/13

double angle formula is sin2x = 2sinxcosx

if sinx=-5/13, then cosx = -12/13

sin2x = 2(-5/13)(-12/13) = 120/169

If x is in the 3rd quadrant, with pi < x < 3pi/2 then

2pi< 2x < 3pi, with 2x in either the 1st or 2nd quadrant, when sin2x>0

Answer 2

In the third quadrant cos x = -12/13 by the Pythagorean theorem.

sin 2x = 2 sin x cos x.

(Note: the sin will be in either quadrant 1 or quadrant 2 where the sin is positive!)