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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 = 3tanA−tan3A1−3tan2A
Now we will proof the above multiple angle formula step-by-step.
Proof: tan 3A
= tan (2A + A)
= tan2A+tanA1−tan2A⋅tanA
= 2tanA1−tan2A+tanA1−2tanA1−tan2A⋅tanA
= 2tanA+tanA−tan3A1−tan2A−2tan2A
= 3tanA−tan3A1−3tan2A
Therefore, tan 3A = 3tanA−tan3A1−3tan2A
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° = 3tan10°−tan310°1−3tan210°.
(ii) The value of tan 3A can also be obtain by putting A = B = C in the formula
tan (A + B + C) = tanA+tanB+tanC−tanAtanBtanC1−tanAtanB−tanBtanC−tanCtanA
11 and 12 Grade Math
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