## Tuesday, 7 December 2021

### Powers of sine and cosine as multiple angles

I had a problem which required integrating powers of sines and cosines.
I used integral-calculator to do the work up to a power of six. Its results are shown below. $$\fbox{\begin{matrix}n&\int{\cos^n{\theta}d\theta}&\int{\sin^n{\theta}d\theta}\\2&\frac{\sin{\left(2\theta\right)}+2\theta}{4}&\frac{-\sin{\left(2\theta\right)}+2\theta}{4}\\3&\sin{\theta}-\frac{\sin^3{\theta}}{3}&\frac{\cos^3{\theta}}{3}-\cos{\theta}\\4&\frac{\sin{\left(4\theta\right)}+8\sin{\left(2\theta\right)}+12\theta}{32}&\frac{\sin{\left(4\theta\right)}-8\sin{\left(2\theta\right)}+12\theta}{32}\\5&\frac{\sin^5{\theta}}{5}-\frac{2\sin^3{\theta}}{3}+\sin{\theta}&-\frac{\cos^5{\theta}}{5}+\frac{2\cos^3{\theta}}{3}-\cos{\theta}\\6&\frac{\sin{\left(6\theta\right)}+9\sin{\left(4\theta\right)}+45\sin{\left(2\theta\right)}+60\theta}{192}&-\frac{\sin{\left(6\theta\right)}-9\sin{\left(4\theta\right)}+45\sin{\left(2\theta\right)}-60\theta}{192}\\\end{matrix}}$$
There appears to be a pattern. But at the moment it is veiled. The integrals of odd powers can be found by a crafty substitution which I found here. It gives$$\int\sin^n{\theta}d\theta=\sum_{l=0}^{l=\frac{n-1}{2}}{\left(-1\right)^{l+1}B_l^{\frac{n-1}{2}}}\frac{\cos^{2l+1}{\theta}}{2l+1}+C,\ for\ odd\ n$$$$\int\cos^n{\theta}d\theta=\sum_{l=0}^{l=\frac{n-1}{2}}{\left(-1\right)^lB_l^{\frac{n-1}{2}}\frac{\sin^{2l+1}{\theta}}{2l+1}}+C,\ for\ odd\ n$$where $B_l^n$ is the binomial coefficient$$B_l^n=\frac{n!}{l!\left(n-l\right)!}$$The integrals of even powers are not so simple. I ventured into multiple angle expressions for $\cos^n{\theta},\sin^n{\theta}$ using the Chebyshev method. Multiple angles are easy integrate. I ended up with the Christmas trees at the top.

If you're interested read it here: Sine and Cosine formulas.pdf (12 pages).