Thread: Deriving an equation using Trig formulas

I'm fairly confident in using the Pythagorean, Sum, Difference, Double Angle and half Angle formulas. The problem is that I struggle with the mental gymnastics to demonstrate sin(3x) = something horrendous, in truth I spend far too much time pursuing blind alleys that look right for far too long.

I can get as far as using the Sum rule to break the Sin(3x) into say

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

But getting it into the final form always seems a step too far.

Can anyone offer some pointers as to how you would go about solving such problems (the only example I have is an assignment question so it would be dishonest of me to post a variation on it for help).

Originally Posted by AliceFisher

I'm fairly confident in using the Pythagorean, Sum, Difference, Double Angle and half Angle formulas. The problem is that I struggle with the mental gymnastics to demonstrate sin(3x) = something horrendous, in truth I spend far too much time pursuing blind alleys that look right for far too long.

I can get as far as using the Sum rule to break the Sin(3x) into say

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

But getting it into the final form always seems a step too far.

Can anyone offer some pointers as to how you would go about solving such problems (the only example I have is an assignment question so it would be dishonest of me to post a variation on it for help).

You're almost done! Remember that you can rewrite as or - it will help in some situations (I haven't used it here, although I thought it was worth mentioning).

.

So multiply through:


Collect like terms:


(see below for explanation of this stage)

Again, multiply through:


Again, collect like terms:


You'll have to check my working for silly errors, but the key thing to apply when you've reached the stage you have is just and rearrange that to convert everything into one trig. function - I used it at the indicated stage to convert the cosine function into a sine function.