Expert: Jack Cheng - 11/28/2006

Question

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The text above is a follow-up to ...

-----Question-----
Find a simplified formula for the expression
(a) sin (3x) in terms of sin(x)
(b) cos (3x) in terms of cos(x)
-----Answer-----
Hi Israel,

For these two problems, you have to use the trig formulas:

sin (A+B) = sin A cos B + sin B cos A
cos (A+B) = cos A cos B - sin A sin B

(There is a special case when A=B; so sin 2A = 2 sin A cos A; cos 2A = cos^2 A - sin^2 A)

We'll have use those formulas twice.

[VIEW BELOW WITH FIXED-SIZE FONT LIKE COURIER.]

sin (3x) = sin 2x cos x + cos 2x sin x
        = 2 sin x cos x (cos x) + (cos^2 x - sin^2 x) sin x
        = 2 sin x cos^2 x + cos^2 x sin x - sin^3 x
        = 3 sin x cos^2 x - sin^3 x

i get this answer that you get to and also for cos i get
cos^3(x)-3sin^2(x)cos(x)

but my teachers answer sheet says
sin= 3sinx-4sin^3x
cos= 4cos^3x-3cos x

could it be that she has the wronge answer or different equation?

Answer
The answers are the same.

 3 sin x cos^2 x - sin^3 x
= 3 sin x (1 - sin^2 x) - sin^3 x
= 3 sin x - 3 sin^3 x - sin^3 x
= 3 sin x - 4 sin^3 x

The same can be done for cos.

Your teacher's answer is better though, because the question does ask you to the answer in terms of sin(x) / cos(x). (I originally misunderstood it to just get rid of the coefficient of x, so I stopped where I did.)

Sorry about my initial misunderstanding.

~ Jack