# Chaitanya's Random Pages

## October 24, 2012

### How I remember the triple angle formulas

Filed under: mathematics — ckrao @ 5:04 am

The benefit of this post is for me to recall better the two following formulas: $\displaystyle \cos 3x = 4\cos^3 x - 3 \cos x$ $\displaystyle \sin 3x = 3\sin x - 4\sin^3 x$

In the past, whenever I have had to find the cosine or sine of three times and angle (e.g. for some maths  contest problems), I have manually performed expansions based on the double angle formulas: \begin{aligned} \cos 3x &= \cos (2x + x) \\&= \cos 2x \cos x - \sin 2x \sin x \\ &= (2 \cos^2 x - 1) \cos x - 2 \sin^2 x \cos x \\ &= (2 \cos^2 x - 1) \cos x - 2(1 - \cos^2 x) \cos x \\&= 4\cos^3 x - 3\cos x \end{aligned} \begin{aligned} \sin 3x &= \sin (2x + x) \\ &= \sin 2x \cos x + \cos 2x \sin x \\&= 2 \sin x \cos^2 x + (1 -2 \sin^2 x) \sin x \\ &= 2\sin x (1 - \sin^2 x ) + (1 - 2\sin^2 s) \sin x \\ &= 3 \sin x - 4 \sin^3 x \end{aligned}

Now that I see these formulas together, I want a way of remembering them without having to write this many lines. Here is my aid.

• Both formulas are of the form $ay^3 + by$, where $y = \cos x$ for the cosine formula and $y = \sin x$ for the sine formula (I already knew this).
• The coefficients up to sign are 4 and 3 (I already knew this).
• The first coefficient is positive, the second negative (hence 4, -3 or 3, -4).
• The cubic term is paired up with 4. Alternatively, the cubic term is not paired up with 3

This last point comes about because the 2 from $2 \sin x \cos x$ is added to the 2 from $2\cos^2 x - 1 = 1 - 2 \sin^2 x$.

If one is unsure whether $\cos 3x$ is $4\cos ^3 x - 3 \cos x$ or $3 \cos x - 4\cos^3 x$, simply substitute $x = 0$ – the expression should evaluate to 1. The coefficients for $\sin 3x$ are then flipped in sign.

## 1 Comment »

1. if n=4k+3, the polynomial for sin(nx) and cos(nx) have opposite sign.
if n=4k+1, they will have the same polynomial form.

Example for n=5. Let f5(x) = 5 x – 20 x^3 + 16 x^5

sin(5 x) = f5(sin(x))
cos(5 x) = f5(cos(x))