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Question
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Options
`63/65`
`62/65`
`61/65`
`60/65`
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Solution
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`
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