Question

9π/8 - 9/4sin⁻¹ ⅓ = 9/4 sin⁻¹ 2√2/3

Answer 1

Given   : 9π/8 - (9/4)sin⁻¹ ⅓ = (9/4) sin⁻¹ (2√2/3)

To find :   सिद्ध कीजिए

Solution:

9π/8 - (9/4)sin⁻¹ ⅓ = (9/4) sin⁻¹ (2√2/3)

LHS =   9π/8 - (9/4)sin⁻¹ ⅓

= (9/4) (π/2 -  sin⁻¹ ⅓)  

Cos⁻¹x + Sin⁻¹x = π/2

=> Cos⁻¹x = π/2 - Sin⁻¹x

=>  Cos⁻¹⅓ = π/2 - Sin⁻¹⅓

= (9/4)  (Cos⁻¹⅓)

Cos⁻¹⅓ = x

=> cosx = 1/3

sin x = √(1 - Cos²x)

=> sin x =  √(1 - (1/3)²)

=> sin x =  √(8/9)

=> sin x = 2√2 /  3

=>  x =  sin⁻¹ (2√2/3)

=> Cos⁻¹⅓ = sin⁻¹ (2√2/3)

LHS = (9/4)  (Cos⁻¹⅓)

=  (9/4) sin⁻¹ (2√2/3)

= RHS

LHS = RHS

=>   9π/8 - (9/4)sin⁻¹ ⅓ = (9/4) sin⁻¹ (2√2/3)

QED

इति सिद्धम

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