Evaluate: sin (tan-1 \({\frac{24}{7}}\))

1.	Evaluate: sin (tan-1 \({\frac{24}{7}}\))
Answer» Given sin as (tan^-1 24/7) Let tan^-1 (24/7) = y tan y = 24/7, where y ∈ [0,π/2] Let us find sin(tan^-1 (24/7)) = sin y As we know that, 1 + cot²θ = cosec²θ 1 + cot² y = cosec² y On, substituting this trigonometry form, we get 1 + (7/24)² = cosec²y 1 + (49/576) = 1/sin²y By rearranging we get, sin²y = 576/625 sin y = 24/25, where y ∈ [0,π/2] sin(tan^-1(24/7)) = 24/25

Answer»

Given sin as (tan^-1 24/7)

Let tan^-1 (24/7) = y

tan y = 24/7, where y ∈ [0,π/2]

Let us find

sin(tan^-1 (24/7)) = sin y

As we know that,

1 + cot²θ = cosec²θ

1 + cot² y = cosec² y

On, substituting this trigonometry form, we get

1 + (7/24)² = cosec²y

1 + (49/576) = 1/sin²y

By rearranging we get,

sin²y = 576/625

sin y = 24/25, where y ∈ [0,π/2]

sin(tan^-1(24/7)) = 24/25

Discussion