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Margarita [4]
4 years ago
15

Write the given expression in terms of x and y only. sin(sin^−1 (x) + cos^−1 (y))

Mathematics
2 answers:
Alex4 years ago
6 0

Answer:

sin(sin^{-1}(x) + cos^{-1}(y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}

Step-by-step explanation:

Given

sin(sin^{-1}(x) + cos^{-1}(y))

Let's define

\alpha = sin^{-1}(x)

\beta = cos^{-1}(y)

Replacing

sin(\alpha + \beta)

sin(\alpha) cos(\beta) + sin(\beta) cos(\alpha)

But

sin(\alpha) = sin(sin^{-1}(x))=x

cos(\beta) = cos(cos^{-1}(y))=y

From trigonometric identity

sin^2(\beta) + cos^2(\beta) = 1

sin(\beta) = \sqrt{1 - cos^2(\beta)} = \sqrt{1 - y^2}

sin^2(\alpha) + cos^2(\alpha) = 1

cos(\alpha) = \sqrt{1 - sin^2(\alpha)} = \sqrt{1- x^2}

Replacing

sin(sin^{-1}(x) + cos^-1 (y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}

pav-90 [236]4 years ago
5 0
We are given the expression sin (sin-1 (X) + cos-1 (y)). We use the associative property to distribute the sine function. sin (sin-1 (x)) is equal to x. cos-1 (y) is equal to beta, the other angle besides alpha. sin beta is also equal to x which means the simplified term is 2x. This makes sense because sin-1 x = cos-1 y 
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