Question

Find an algebraic expression equivalent the given expression to sin(cos-1(x))

Answer 1

this is simply a quick addition to the superb reply above by "MastG"

`\textit{Pythagorean Identities} \\\\ \sin^2(\theta)+\cos^2(\theta)=1\implies \sin^2(\theta)=1-\cos^2(\theta)\implies \sin(\theta)=\boxed{√(1-\cos^2(\theta))} \\\\[-0.35em] \rule{34em}{0.25pt}`

`sin( ~~ \underset{\theta }{cos^(-1)(x)} ~~ )\implies sin(\theta ) \\\\[-0.35em] ~\dotfill\\\\ cos^(-1)(x)=\theta \implies cos(\theta )=x\implies cos^2(\theta )=x^2 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{so then we can say that}}{sin( ~~ cos^(-1)(x) ~~ )\implies sin(\theta )}\implies √(1-\cos^2(\theta))\implies \boxed{√(1-x^2)}`

Answer 2

Answer: √(1 - x^2)

Step-by-step explanation: The inverse cosine function, cos^-1(x), gives the angle whose cosine is x. So, cos^-1(x) = θ where cos(θ) = x.Using the identity sin(cos^-1(x)) = √(1 - x^2), we can simplify the expression sin(cos^-1(x)) as:sin(cos^-1(x)) = √(1 - x^2)So, the algebraic expression equivalent to sin(cos^-1(x)) is √(1 - x^2).