We will learn how to express the multiple angle of tan 3A in terms of A or tan 3A in terms of tan A.
Trigonometric function of tan 3A in terms of tan A is also known as one of the double angle formula.
If A is a number or angle then we have, tan 3A = \(\frac{3 tan A - tan^{3} A}{1 - 3 tan^{2} A}\)
Now we will proof the above multiple angle formula step-by-step.
Proof: tan 3A
= tan (2A + A)
= \(\frac{tan 2A + tan A}{1 - tan 2A \cdot tan A}\)
= \(\frac{\frac{2 tan A}{1 - tan^{2} A} + tan A}{1 - \frac{2 tan A}{1 - tan^{2} A} \cdot tan A}\)
= \(\frac{2 tan A + tan A - tan^{3} A}{1 - tan^{2} A - 2 tan^{2} A}\)
= \(\frac{3 tan A - tan^{3} A}{1 - 3 tan^{2} A}\)
Therefore, tan 3A = \(\frac{3 tan A - tan^{3} A}{1 - 3 tan^{2} A}\)
Note:
(i) In the above formula we should note that the angle on the R.H.S. of the formula is one-third of the angle on L.H.S. Therefore, tan 30° = \(\frac{3 tan 10° - tan^{3} 10°}{1 - 3 tan^{2} 10°}\).
(ii) The value of tan 3A can also be obtain by putting A = B = C in the formula
tan (A + B + C) = \(\frac{tan A + tan B + tan C - tan A tan B tan C}{1 - tan A tan B - tan B tan C - tan C tan A}\)
11 and 12 Grade Math
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