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  \(\frac{sin A}{cos A}\) ∙ cos\(^{2}\) A

⇒ sin 2A = 2 tan A ∙  \(\frac{1}{sec^{2} A}\)

⇒ sin 2A = \(\frac{2  tan A}{1 + tan^{2} A}\)

There for sin 2A =  \(\frac{2  tan A}{1 + tan^{2} A}\)

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)


Given, sin 2A = 4/5  

Therefore, \(\frac{2  tan A}{1  +  tan^{2} A}\) = 4/5

 ⇒ 4 + 4 tan\(^{2}\) A = 10 tan A            

⇒ 4 tan\(^{2}\) A - 10 tan A + 4 = 0

⇒ 2 tan\(^{2}\) A - 5 tan A + 2 = 0

⇒ 2 tan\(^{2}\) 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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