Hi,

I hope someone can help.

I'm suppose to prove that sin(2pi/3) = sin(8pi/3) based on the formula that I developed: Sin4x = 4SinxCos-8Sin^3Cosx

This is the textbook solution for the question:

I believe what the solution is trying to say is that we already know sin(2pi/3) since (2pi/3) is a common angle on the unit circle (hence why there was only one step in the textbook solution to know what the ratio is for sin(2pi/3).

Now that I said what I understand, I will now explain what I'm confused about...

1.Can someone please explain to me why 2pi/3 is subbed into the Sin4x equation as x? What is the textbook solution trying to communicate by doing that? I understand how one could sub 2pi/3 into Sin4x as x, but I'm not sure why you would in the first place. Why don't you just sub in 8pi/3 as x?

2.In addition, do you even need to have the equation of Sin4x in order to determine that sin(2pi/3) and sin(8pi/3) are equivalent? For example, I know that ((8 x 180)/3)=480... and 480 - 360 = 120... and 180 - 120 = 60. What I just proved is that the angle 8pi/3 lies within the 2nd quadrant and has a related acute angle of 60 degrees... which is the same as 2pi/3! Hopefully what I just said didn't sound too convoluted...

Sincerely,

Olivia