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sin 2A in Terms of tan A

We will learn how to express the multiple angle of sin 2A in terms of tan A.

Trigonometric function of sin 2A in terms of tan A is also known as one of the double angle formula.

We know if A is a number or angle then we have,

sin 2A = 2 sin A cos A

⇒ sin 2A = 2  sinAcosA ∙ cos2 A

⇒ sin 2A = 2 tan A ∙  1sec2A

⇒ sin 2A = 2tanA1+tan2A

There for sin 2A =  2tanA1+tan2A

Now, we will apply the formula of multiple angle of sin 2A in terms of tan A to solve the below problem.

1. If sin 2A = 4/5 find the value of tan A (0 ≤ A ≤ π / 4)

Solution:

Given, sin 2A = 4/5  

Therefore, 2tanA1+tan2A = 4/5

 ⇒ 4 + 4 tan2 A = 10 tan A            

⇒ 4 tan2 A - 10 tan A + 4 = 0

⇒ 2 tan2 A - 5 tan A + 2 = 0

⇒ 2 tan2 A - 4 tan A - tan A + 2 = 0

⇒ 2 tan A (tan A - 2) - 1 (tan A - 2) =0

⇒ (tan A - 2) (2 tan A - 1) = 0

Therefore, tan A - 2 = 0 and 2 tan A - 1 = 0

⇒ tan A = 2 and tan A = 1/2

According to the problem, 0 ≤ A ≤ π/4     

Therefore, tan A = 2 is impossible

Therefore, the required value of tan A is 1/2.

 Multiple Angles






11 and 12 Grade Math

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