Question

Evaluate: tan(1/2 sin^-1(3/5))
Plz fast..

Answer 1

Answer:

Evaluate: tan(1/2 sin^-1(3/5))

Answer 2

The value of  tan( 1/2 sin^-1 (3/5)) is 1/3 , - 3.

Given: tan ( 1/2 sin^-1 (3/5))

To Find: Evaluate tan( 1/2 sin^-1 (3/5))

Solution:

Let us assume sin^-1 (3/5) = θ                  ....... (1)

∴ sin θ = 3/5

By triangle law, we see that,

tan θ = 3/4                                                    ...... (2)

From the actual expression, tan(1/2 sin^-1(3/5)), when we substitute sin^-1 (3/5) = θ, we get the reduced expression as,

⇒ tan ( θ/2 )

Now, we can solve it by using the trigonometric formula,

            tan θ =  2 × tan (θ/2) / ( 1 - tan²(θ/2))

        ⇒ 3/4 = 2 × tan (θ/2) / ( 1 - tan²(θ/2))              [ as tan θ  = 3/4 ]

        ⇒ 3 × (1 - tan²(θ/2)) = 8 × tan (θ/2)

        ⇒ 3 tan²(θ/2) + 8 tan (θ/2) - 3 = 0

        ⇒ 3 tan²(θ/2) + 9 tan (θ/2) - tan (θ/2) - 3 = 0

        ⇒ 3 tan (θ/2) [ tan (θ/2) + 3 ] - 1 [ tan (θ/2) + 3 ] = 0

        ⇒ [ 3 tan (θ/2) - 1 ] [ tan (θ/2) + 3 ] = 0

        ⇒ tan (θ/2) = 1/3 , - 3

Hence, the value of  tan( 1/2 sin^-1 (3/5)) is 1/3 , - 3.

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