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.

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