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.






11 and 12 Grade Math

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