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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