# Powers of Trigonometric Functions

Angle: $$\alpha$$
Trigonometric functions: $$\sin \alpha$$, $$\cos \alpha$$, $$\tan \alpha$$
1. Sine squared
$${\sin^2}\alpha =$$ $$\large\frac{{1 – \cos 2\alpha }}{2}\normalsize$$
2. Sine cubed
$${\sin^3}\alpha =$$ $$\large\frac{{3\sin \alpha – \sin 3\alpha }}{4}\normalsize$$
3. Sine to the fourth power
$${\sin^4}\alpha =$$ $$\large\frac{{\cos 4\alpha – 4\cos 2\alpha + 3}}{8}\normalsize$$
4. Sine to the fifth power
$${\sin^5}\alpha =$$ $$\large\frac{{10\sin \alpha – 5\sin 3\alpha + \sin 5\alpha}}{16}\normalsize$$
5. Sine to the sixth power
$${\sin^6}\alpha =$$ $$\large\frac{{10 – 15\cos 2\alpha + 6\cos 4\alpha – \cos 6\alpha}}{32}\normalsize$$
6. Cosine squared
$${\cos^2}\alpha =$$ $$\large\frac{{1 + \cos 2\alpha }}{2}\normalsize$$
7. Cosine cubed
$${\cos^3}\alpha =$$ $$\large\frac{{3\cos \alpha + \cos 3\alpha }}{4}\normalsize$$
8. Cosine to the fourth power
$${\cos^4}\alpha =$$ $$\large\frac{{\cos 4\alpha + 4\cos 2\alpha + 3}}{8}\normalsize$$
9. Cosine to the fifth power
$${\cos^5}\alpha =$$ $$\large\frac{{10\cos \alpha + 5\sin 3\alpha + \cos 5\alpha}}{16}\normalsize$$
10. Cosine to the sixth power
$${\cos^6}\alpha =$$ $$\large\frac{{10 + 15\cos 2\alpha + 6\cos 4\alpha + \cos 6\alpha}}{32}\normalsize$$
11. Tangent squared
$${\tan^2}\alpha = {\large\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha}}\normalsize} =$$ $${\large\frac{{1 – \cos 2\alpha }}{{1 + \cos 2\alpha }}\normalsize}$$
12. Tangent cubed
$${\tan^3}\alpha = {\large\frac{{{{\sin }^3}\alpha }}{{{{\cos }^3}\alpha}}\normalsize} =$$ $${\large\frac{{3\sin \alpha – \sin 3\alpha }}{{3\cos \alpha + \cos 3\alpha}}\normalsize}$$