Get the answer to "Express cos⁡(sin−1⁡x−cos−1y) as an algebraic expression in x..."

College
Algebra
College algebra
Upper level algebra

lwfrgin

Answered question

2020-10-27

Express

\cos (\sin^{- 1} x - c o s^{- 1} y)

as an algebraic expression in x and y.

Answer & Explanation

liingliing8

Skilled2020-10-28Added 95 answers

Step1
Consider the given expression as $\cos (\sin^{- 1} x - \cos^{- 1} y)$ .
$L e t \sin^{- 1} x = α a n d \cos^{- 1} y = β, t h e n x = \sin α a n d y = \cos β$ .
Now the given expression can be write as $\cos (α — β)$ .
Known formula:
l. $\cos (α — β) = \cos α \cos β + \sin α \sin β$
2. $\sin^{2} 0 + \cos^{2} 0 = 1$
Step 2
Compute the value of $\cos α a n d \sin β$ as follows.
$\cos α = \sqrt{1 - \sin^{2} α}$
$= \sqrt{1 - x^{2}}$
$\sin β = \sqrt{1 - \cos^{2} β}$
$= \sqrt{1 - y^{2}}$
Substitute the values of $\cos α a n d \sin β$ in the formula (1).
$\cos (α — β) = \cos α \cos β + \sin α \sin β$
$= (\sqrt{1 - x^{2}}) (y) + (x) (\sqrt{1 - y^{2}})$
$= x \sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}$
Thus, the expression $\cos (\sin^{- 1} x - \cos^{- 1} y)$ as algebraic in x and y as x $\sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}$

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