# Submultiple Angle Formulae

The important trigonometrical ratios of submultiple angle formulae are given below:

(i) sin A = 2 sin $$\frac{A}{2}$$ cos $$\frac{A}{2}$$

(ii) cos A = cos$$^{2}$$ $$\frac{A}{2}$$ – sin$$^{2}$$ $$\frac{A}{2}$$

(iii) cos  A = 2 cos$$^{2}$$ $$\frac{A}{2}$$ - 1

(iv) cos A = 1 - 2 sin$$^{2}$$ $$\frac{A}{2}$$

(v) 1 + cos A = 2 cos$$^{2}$$ $$\frac{A}{2}$$

(vi) 1 - cos A = 2 sin$$^{2}$$ $$\frac{A}{2}$$

(vii) tan$$^{2}$$ $$\frac{A}{2}$$ = $$\frac{1 - cos A}{1 + cos A}$$

(viii) sin A = $$\frac{2 tan \frac{A}{2}}{1 + tan^{2} \frac{A}{2}}$$

(ix) cos A = $$\frac{1 - tan^{2} \frac{A}{2}}{1 + tan^{2} \frac{A}{2}}$$

(x) tan A = $$\frac{2 tan \frac{A}{2}}{1 - tan^{2} \frac{A}{2}}$$

(xi) sin A = 3 sin $$\frac{A}{3}$$ - 4 sin$$^{3}$$ $$\frac{A}{3}$$

(xii) cos A = 4 cos$$^{3}$$ $$\frac{A}{3}$$ - 3 cos $$\frac{A}{3}$$

(xiii) sin 15° = cos 75° = $$\frac{√3 - 1}{2√2}$$

(xiv) cos 15° = sin 75° = $$\frac{√3 + 1}{2√2}$$

(xv) tan 15° = 2 - √3.

(xvii) sin 22½˚ = $$\frac{1}{2}\sqrt{2 - \sqrt{2}}$$

(xvii) cos 22½˚ = $$\frac{1}{2}\sqrt{2 - \sqrt{2}}$$

(xviii) tan 22½˚= √2 - 1.

(xix) sin 18 ° = cos 72° = $$\frac{√5 - 1}{4}$$

(xx) cos 18° = sin 72° = $$\frac{\sqrt{10 + 2\sqrt{5}}}{4}$$

(xxi) cos 36° = cos 72° = $$\frac{√5 + 1}{4}$$

(xxii) sin 36° = cos 54° = $$\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$