tan 2A in Terms of A

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  2tanA1tan2A?

We know that for two real numbers or angles A and B,

tan (A + B) =  tanA+tanB1tanAtanB

Now, putting B = A on both sides of the above formula we get,

tan (A + A) =  tanA+tanA1tanAtanA

⇒ tan 2A =  2tanA1tan2A

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°1tan230°.

(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)

= 2tan2A1tan2(2A), [Since we know 2tanA1tan2A]

= 22tanA1tan2A1(2tanA1tan2A)2

= 4tanA(1tan2A)(1tan2A)24tan2A

= 4tanA(1tan2A)16tan2A+4tan4

 Multiple Angles







11 and 12 Grade Math

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