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

11 and 12 Grade Math

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