How do I expand cos nθ and sin nθ in powers of cos θ and sin θ?

In one line

Write \(\cos n\theta+i\sin n\theta=(\cos\theta+i\sin\theta)^n\), expand the right side by the binomial theorem, and compare real and imaginary parts.

Explained simply

You already know \(\cos2\theta=\cos^2\theta-\sin^2\theta\) and \(\sin2\theta=2\sin\theta\cos\theta\). This method produces the same kind of formula for \(3\theta,4\theta,5\theta,\dots\) without memorising anything: the binomial expansion does the work and the real part gives cosine, the imaginary part gives sine.

Technically, with \(c=\cos\theta\), \(s=\sin\theta\): \(\cos n\theta+i\sin n\theta=(c+is)^n=\sum_{r=0}^{n}{}^nC_r\,c^{n-r}(is)^r\). Since \(i^2=-1\), \(i^3=-i\), \(i^4=1\), even \(r\) terms are real and odd \(r\) terms are imaginary. Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal, which is what lets us split one equation into two formulas.

How it works

1. Write \(\cos n\theta+i\sin n\theta=(c+is)^n\).
2. Expand with binomial coefficients \({}^nC_r\).
3. Simplify powers of \(i\) and separate real and imaginary terms.
4. Equate: real part \(=\cos n\theta\), imaginary part \(=\sin n\theta\).
5. If asked "in powers of \(\cos\theta\) only", replace \(s^2=1-c^2\).
6. For \(\tan n\theta\), divide \(\sin n\theta\) by \(\cos n\theta\), then divide top and bottom by \(c^n\).

Key formula

\(\cos n\theta=c^n-{}^nC_2c^{n-2}s^2+{}^nC_4c^{n-4}s^4-\dots\)

\(\sin n\theta={}^nC_1c^{n-1}s-{}^nC_3c^{n-3}s^3+\dots\)

\(\tan n\theta=\dfrac{{}^nC_1t-{}^nC_3t^3+{}^nC_5t^5-\dots}{1-{}^nC_2t^2+{}^nC_4t^4-\dots}\), where \(t=\tan\theta\).

Quick example

Expand \(\cos4\theta\) and \(\sin4\theta\).

\((c+is)^4=c^4+4c^3(is)+6c^2(is)^2+4c(is)^3+(is)^4=c^4+4ic^3s-6c^2s^2-4ics^3+s^4\).

Real part: \(\cos4\theta=c^4-6c^2s^2+s^4\). Imaginary part: \(\sin4\theta=4c^3s-4cs^3\).

In powers of \(\cos\theta\) only: \(\cos4\theta=c^4-6c^2(1-c^2)+(1-c^2)^2=8c^4-8c^2+1\). Check at \(\theta=0\): \(8-8+1=1=\cos0\).

Common mistakes

• Getting the sign of \(i^3=-i\) or \(i^4=+1\) wrong; write the powers of \(i\) explicitly before collecting terms.
• Forgetting binomial coefficients and writing \((c+is)^4=c^4+(is)^4\).
• Not checking the result at \(\theta=0\) or \(\theta=\pi/2\), which catches most sign errors.

How MU asks it

Typical 5/6-mark parts: "Expand \(\cos7\theta\) in powers of \(\cos\theta\)", "Prove that \(\dfrac{\sin6\theta}{\sin\theta}=32c^5-32c^3+6c\)", or "Express \(\tan5\theta\) in terms of \(\tan\theta\)". For full marks: state De Moivre's theorem, write the full binomial expansion with coefficients, show the equating of real/imaginary parts, then the substitution \(s^2=1-c^2\) step by step.