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

Answer 1

recall the fun trig identity:

cos(2x)=cos(x)^2-sin(x)^2

=(1-sin(x)^2)-sin(x)^2

=1-2sin(x)^2

=1-2(sin x)^2

given that sin x=-5/13, this implies that

cos(2x)=1-2(-5/13)^2

=1-2(25/169)

=1-50/169

=(169-50)/169

=119/169

thus cos (2x)= 119/169