Sub-multiple angles

• Sub-multiple angles are formed by dividing an angle by a positive integer.
• The trigonometric functions of sub-multiple angles can be expressed in terms of the trigonometric functions of the original angle.
• The sub-multiple angle formulas are used to simplify trigonometric expressions and solve trigonometric equations.
• The most commonly used sub-multiple angle formulas are the half-angle and quarter-angle formulas, which can be used to find the values of trigonometric functions for angles that are half or a quarter the size of known angles.
• The sub-multiple angle formulas can also be used to find the values of trigonometric functions for angles that are one-third or one-fifth the size of known angles.

Important Formulas to Remember in these Sub-multiple angles

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• $\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}$
• $\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}$
• $\tan \frac{x}{2}=\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}$