# tan 3A in Terms of A

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) = $$\frac{tan A + tan B + tan C - tan A tan B tan C}{1 - tan A tan B - tan B tan C - tan C tan A}$$

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