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

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

(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.

11 and 12 Grade Math

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