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° = \(\frac{\sqrt{10  -  2\sqrt{5}}}{4}\)

● Submultiple Angles






11 and 12 Grade Math

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