We will learn to express trigonometric function of tan 2A in terms of A or tan 2A in terms of tan A. We know if A is a given angle then 2A is known as multiple angles.
How to proof the formula of tan 2A is equals 2tanA1−tan2A?
We know that for two real numbers or angles A and B,
tan (A + B) = tanA+tanB1−tanAtanB
Now, putting B = A on both sides of the above formula we get,
tan (A + A) = tanA+tanA1−tanAtanA
⇒ tan 2A = 2tanA1−tan2A
Note: (i) In the above formula we should note that the angle on the R.H.S. is half of the angle on L.H.S. Therefore, tan 60° = 2tan30°1−tan230°.
(ii) The above formula is also known as double angle formulae for tan 2A.
Now, we will apply the formula of multiple angle of tan 2A in terms of A or tan 2A in terms of tan A to solve the below problem.
1. Express tan 4A in terms of tan A
Solution:
tan 4a
= tan (2 ∙ 2A)
= 2tan2A1−tan2(2A), [Since we know 2tanA1−tan2A]
= 2⋅2tanA1−tan2A1−(2tanA1−tan2A)2
= 4tanA(1−tan2A)(1−tan2A)2−4tan2A
= 4tanA(1−tan2A)1−6tan2A+4tan4
11 and 12 Grade Math
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