# Submultiple angle formulas

1

### Half angle formulas

$(1)\,\,\,\,$ $\sin \theta = 2\sin \Bigg(\dfrac{\theta}{2}\Bigg) \cos \Bigg(\dfrac{\theta}{2}\Bigg) = \dfrac{2 \tan \Bigg(\dfrac{\theta}{2}\Bigg)}{1+\tan^2 \Bigg(\dfrac{\theta}{2}\Bigg)}$

$(2)\,\,\,\,$ $\cos \theta = \cos^2 \Bigg(\dfrac{\theta}{2}\Bigg) -\sin^2 \Bigg(\dfrac{\theta}{2}\Bigg) = 2\cos^2 \Bigg(\dfrac{\theta}{2}\Bigg) -1 = 1 -2\sin^2 \Bigg(\dfrac{\theta}{2}\Bigg)$

$(3)\,\,\,\,$ $\tan \theta = \dfrac{2 \tan \Bigg(\dfrac{\theta}{2}\Bigg)}{1-\tan^2 \Bigg(\dfrac{\theta}{2}\Bigg)} = \dfrac{2}{\cot \Bigg(\dfrac{\theta}{2}\Bigg) -\tan \Bigg(\dfrac{\theta}{2}\Bigg)}$

$(4)\,\,\,\,$ $\cot \theta = \dfrac{\cot^2 \Bigg(\dfrac{\theta}{2}\Bigg) -1}{2 \cot \Bigg(\dfrac{\theta}{2}\Bigg)} = \dfrac{\cot \Bigg(\dfrac{\theta}{2}\Bigg) -\tan \Bigg(\dfrac{\theta}{2}\Bigg)}{2}$

2

### One third angle formulas

$(1)\,\,\,\,$ $\sin \theta = 3\sin \Bigg(\dfrac{\theta}{3}\Bigg) -4\sin^3 \Bigg(\dfrac{\theta}{3}\Bigg)$

$(2)\,\,\,\,$ $\cos \theta = 4\cos^3 \Bigg(\dfrac{\theta}{3}\Bigg) -3\cos \Bigg(\dfrac{\theta}{3}\Bigg)$

$(3)\,\,\,\,$ $\tan \theta = \dfrac{3\tan \Bigg(\dfrac{\theta}{3}\Bigg) -tan^3 \Bigg(\dfrac{\theta}{3}\Bigg)}{1 -3tan^2 \Bigg(\dfrac{\theta}{3}\Bigg)}$

$(4)\,\,\,\,$ $\cot \theta = \dfrac{3\cot \Bigg(\dfrac{\theta}{3}\Bigg) -\cot^3 \Bigg(\dfrac{\theta}{3}\Bigg)}{1 -3\cot^2 \Bigg(\dfrac{\theta}{3}\Bigg)}$