# Thread: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent ratios?

1. ## Complex numbers (I'm sure), trigonometric identities (I think) and equivalent ratios?

given that cis 3Ө = (cos3 Ө - 3 cos Ө sin2 Ө) + i (3 cos2 Ө sin Ө - sin3 Ө) it follows that cos 3Ө is equal to?

I really don't know how to do this, any help would be appreciated :/

2. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

cis $3\theta$ is shorthand for $\cos 3\theta + i\sin 3\theta$ so your first step is to equate reals and imaginaries and to see what that leads to.

3. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

I did try cos 3Ө = cos^3 Ө - 3 cos Ө sin^2 Ө
but I don't know what to do after that point. I went to this site Table of Trigonometric Identities for trig identities and subbed them in, but to no avail.

4. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

How about making use of the identity $\cos^{2}\theta + \sin^{2}\theta = 1$ ?

5. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

oh my, I think I may have actually got it thanks to that ^
but not sure if my working is legit or not.

One question,
is there an equivalent for cos^3 (Ө), or any with powers?
And, would something like cos 3Ө = 3 cos Ө?

6. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

Originally Posted by iamapineapple
One question,
is there an equivalent for cos^3 (Ө), or any with powers?
And, would something like cos 3Ө = 3 cos Ө?

NO, NO . $\cos(3\theta)\ne 3\cos(\theta)$. Learn that!

The are multiple ways to express $\cos^3(\theta)$.
Which one we use depends on the need.

7. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

Yeah, so my working was not legit hahaha Can you help please Plato?! But thanks for answering my question.

8. ## Re: Complex numbers (I'm sure), trigonometric identities (I think) and equivalent rat

Originally Posted by iamapineapple