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Question

What is the formula for tan3A?


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Solution

Find the formula for tan3A.

In the trigonometric formula tan(C+D)=tanC+tanD1-tanCtanD,

Put C=A,D=2A:

tanA+2A=tanA+tan2A1-tanAtan2AtanA+2A=tanA+2tanA1-tan2A1-tanA2tanA1-tan2Atan2A=2tanA1-tan2Atan3A=tanA-tan3A+2tanA1-tan2A1-tan2A-2tan2A1-tan2Atan3A=3tanA-tan3A1-3tan2Atan3A=3tanA-tan3A1-3tan2A

Hence, the formula is tan(3A)=3tanA-tan3A1-3tan2A.


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