# Trigonometric Ratios of Angle $$\frac{A}{2}$$

We will learn about the trigonometric ratios of angle $$\frac{A}{2}$$ in terms of angle A.

How to express sin A, cos A and tan A in terms of $$\frac{A}{2}$$?

(i) For all values of the angle A we know that, sin 2A = 2 sin A cos A

Now replacing A by $$\frac{A}{2}$$ in the above relation then we obtain the relation as,

sin A = 2 sin $$\frac{A}{2}$$ cos $$\frac{A}{2}$$

(ii) For all values of the angle A we know that, cos 2A = cos$$^{2}$$ A – sin$$^{2}$$ A

Now replacing A by $$\frac{A}{2}$$ in the above relation then we obtain the relation as,

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

(iii) For all values of the angle A we know that, cos 2A = 2 cos$$^{2}$$ A - 1 or 1 + cos 2A = 2 cos$$^{2}$$ A

Now replacing A by $$\frac{A}{2}$$ in the above relation then we obtain the relation as,

cos  A = 2 cos$$^{2}$$ $$\frac{A}{2}$$ - 1 or 1 + cos A = 2 cos$$^{2}$$ $$\frac{A}{2}$$

(iv) For all values of the angle A we know that, cos 2A = 1 - 2 sin$$^{2}$$ A or 1 - cos 2A = 2 sin$$^{2}$$ A

Now replacing A by $$\frac{A}{2}$$ in the above relation then we obtain the relation as,

cos A = 1 - 2 sin$$^{2}$$ $$\frac{A}{2}$$ or 1 - cos A = 2 sin$$^{2}$$ $$\frac{A}{2}$$

(v) For all values of the angle A we know that, tan 2A = 2 tan A/1 – tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

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

(vi) For all values of the angle A we know that, sin 2A = 2 tan A/1 + tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

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

(vii) For all values of the angle A we know that, cos 2A = 1 - tan^2 A /1 + tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

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

Note: Formulas of trigonometric ratios of angle A in terms of angle $$\frac{A}{2}$$ is also known as sub-multiple angle.