Question

Sin ' 3/5 +sin 8/17 = COS'36/85​

Answer 1

EXPLANATION.

⇒ sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85).

As we know that,

Formula of :

⇒ cosθ = √1 - sin²θ.

Using this formula in equation, we get.

⇒ sin⁻¹(3/5) = x.

⇒ sin⁻¹(8/17) = y.

⇒ cos x = √1 - sin²x.

⇒ cos x = √1 - (3/5)².

⇒ cos x = √1 - 9/25.

⇒ cos x = √25 - 9/25.

⇒ cos x = √16/25.

⇒ cos x = 4/5.

⇒ cos y = √1 - sin²y.

⇒ cos y = √1 - (8/17)².

⇒ cos y = √1 - 64/289.

⇒ cos y = √289 - 64/289.

⇒ cos y = √225/289.

⇒ cos y = 15/17.

As we know that,

Formula of :

⇒ cos(x + y) = cos(x).cos(y) - sin(x).sin(y).

Using this formula in equation, we get.

⇒ cos(x + y) = 4/5 x 15/17 - 3/5 x 8/17.

⇒ cos(x + y) = 60/85 - 24/85.

⇒ cos(x + y) = 36/85.

⇒ x + y = cos⁻¹(36/85).

Put the value of x and y in equation, we get.

⇒ sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85).

                                                                                                                         

MORE INFORMATION.

Properties.

(1) = sin⁻¹x + cos⁻¹x = π/2.

(2) = tan⁻¹x + cot⁻¹x = π/2.

(3) = sec⁻¹x + cosec⁻¹x = π/2.

Answer 2

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⇒ sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85).

As we know that,

Formula of :

⇒ cosθ = √1 - sin²θ.

Using this formula in equation, we get.

⇒ sin⁻¹(3/5) = x.

⇒ sin⁻¹(8/17) = y.

⇒ cos x = √1 - sin²x.

⇒ cos x = √1 - (3/5)².

⇒ cos x = √1 - 9/25.

⇒ cos x = √25 - 9/25.

⇒ cos x = √16/25.

⇒ cos x = 4/5.

⇒ cos y = √1 - sin²y.

⇒ cos y = √1 - (8/17)².

⇒ cos y = √1 - 64/289.

⇒ cos y = √289 - 64/289.

⇒ cos y = √225/289.

⇒ cos y = 15/17.

As we know that,

Formula of :

⇒ cos(x + y) = cos(x).cos(y) - sin(x).sin(y).

Using this formula in equation, we get.

⇒ cos(x + y) = 4/5 x 15/17 - 3/5 x 8/17.

⇒ cos(x + y) = 60/85 - 24/85.

⇒ cos(x + y) = 36/85.

⇒ x + y = cos⁻¹(36/85).

Put the value of x and y in equation, we get.

⇒ sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85).

                                                                                                                         

MORE INFORMATION.

Properties.

(1) = sin⁻¹x + cos⁻¹x = π/2.

(2) = tan⁻¹x + cot⁻¹x = π/2.

(3) = sec⁻¹x + cosec⁻¹x = π/2.

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