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