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

• Trigonometric Ratios of Angle $\frac{A}{2}$
• Trigonometric Ratios of Angle $\frac{A}{3}$
• Trigonometric Ratios of Angle $\frac{A}{2}$ in Terms of cos A
• tan $\frac{A}{2}$ in Terms of tan A
• Exact value of sin 7½°
• Exact value of cos 7½°
• Exact value of tan 7½°
• Exact Value of cot 7½°
• Exact Value of tan 11¼°
  
• Exact Value of sin 15°
• Exact Value of cos 15°
• Exact Value of tan 15°
  
• Exact Value of sin 18°
• Exact Value of cos 18°
• Exact Value of sin 22½°
• Exact Value of cos 22½°
• Exact Value of tan 22½°
• Exact Value of sin 27°
• Exact Value of cos 27°
• Exact Value of tan 27°
  
• Exact Value of sin 36°
• Exact Value of cos 36°
• Exact Value of sin 54°
• Exact Value of cos 54°
• Exact Value of tan 54°
• Exact Value of sin 72°
• Exact Value of cos 72°
• Exact Value of tan 72°
• Exact Value of tan 142½°
  
• Submultiple Angle Formulae
  
• Problems on Submultiple Angles

11 and 12 Grade Math

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