Solve for exact values.

Question

What are the exact values for sin 4x, cos 4x and tan 4x if you know that tan 2x = -3/2

Ive been stuck on this for ages! Thanks for helping.

Answer 1

sin(4x)=2sin(2x)cos(2x) and cos(4x)=(cos(2x))^2-(sin(2x))^2)=1-2(sin(2x))^2=2(cos(2x))^2-1; these are trig identities true for all x. tan(4x)=sin(4x)/cos(4x). If tan(2x)=-3/2, then using a right-angled triangle we can find the hypotenuse=sqrt(2^2+3^2)=sqrt(13). So sin(2x)=-3/sqrt(13) and cos(2x)=2/sqrt(13).

sin(4x)=-2*2*3/13=-12/13; cos(4x)=4/13-9/13=-5/13; tan(4x)=12/5=2.4.