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

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