# Trigonometric Ratios of Angle $$\frac{A}{3}$$

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

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