Question

If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.

Options

• `63/65`

• `62/65`

• `61/65`

• `60/65`

Answer 1

If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha, "then"  α = bbunderline(63/65)`.

Explanation:

Given: `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`

Step 1: Use identity for inverse cosine

`cos^-1 x = sin^-1 sqrt(1 - x^2)`

`cos^-1 (12/13) = sin^-1 (sqrt(1 - (12/13)^2)) = sin^-1 (5/13)`

Step 2: Now add the two inverse sines

`sin^-1 (4/5) + sin^-1 (5/13)`

`sin^-1 x + sin^-1 y = sin^-1 (xsqrt(1 - y^2) + ysqrt(1 - x^2)), "when"  x^2 + y^2 ≤ 1`

Let x = `4/5, y = 5/13`

Now compute:

`sqrt(1 - y^2) = sqrt(1 - (5/13)^2) = sqrt(144/169) = 12/13`

`sqrt(1 - x^2) = sqrt(1 - (4/5)^2) = sqrt(9/25) = 3/5`

Now plug into the formula:

`alpha = 4/5 * 12/3 + 5/13 * 3/5`

`alpha = 48/65 + 15/65 = 63/65`

`alpha = 63/65`