Given cosθ = − 4/3 and 𝜋 < 𝜃 < 3𝜋/2 Find the exact value of sin 2𝜃

Answer 1

It's quadrant III, where sines <0

cosx = -4/3? You have a typo or mistake. cosines are never less than -1

Maybe you meant to write cosx =-3/4? IF so, -3/4 = adjacent side over hypotenuse with adjacent side =-3 and hypotenuse = 4. then sinx = opposite side over hypotenuse. It's a right triangle, where opposite side squared = hypotenuse squared minus adjacent side squared. o^2= h^2 - a^2 o^2= 4^2 -3^2 = 16-9 = 7. opposite side = square root of 7. sinx= -sqr7/4

sin2x= 2sinxcosx = -2(-sqr7/4)(-3/4)= -(3/8)sqr7= about -0.992 = about- 1

cosx =-3/4.

use a calculator with a trig function. find the inverse cosine of -3/4. x = 138.6 degrees.

sin2x = sin(2(138.6)) = sin277.2 = -0.992 about -1

but if you want the exact value, it's -(3/8)sqr7, slightly >-1

sin(180+90) = sin270 = exactly -1. 277.2 is in quadrant IV.