We will learn about the trigonometric ratios of angle \(\frac{A}{3}\) in terms of angle A.

How to express sin A, cos A and tan A in terms of \(\frac{A}{3}\)?

(i) For all values of the angle A we know that, sin 3A = 3 sin A - 4 sin\(^{3}\) A

Now replacing A by \(\frac{A}{3}\) in the above relation then we obtain the relation as,

sin A = 3 sin \(\frac{A}{3}\) - 4 sin\(^{3}\) \(\frac{A}{3}\)

(ii) For all values of the angle A we know that, cos 3A= 4 cos\(^{3}\) A - 3 cos A

Now replacing A by \(\frac{A}{3}\) in the above relation then we obtain the relation as,

cos A = 4 cos\(^{3}\) \(\frac{A}{3}\) - 3 cos \(\frac{A}{3}\)

(iii) For all values of the angle A we know that, tan 3A = \(\frac{3 tan A - tan^{3} A}{1 - 3 tan^{2} A}\)

Now replacing A by \(\frac{A}{3}\) in the above relation then we obtain the relation as,

tan A = \(\frac{3 tan \frac{A}{3} - tan^{3} \frac{A}{3}}{1 - 3 tan^{2} \frac{A}{3}}\)

**11 and 12 Grade Math**

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