Final answer:
The algebraic expression for cos(sin⁻¹(x)) is √(1 - x²), obtained by considering a right triangle where one angle θ has a sine of x and using the Pythagorean Theorem to find the length of the adjacent side.
Step-by-step explanation:
To rewrite the expression cos(sin⁻¹(x)) as an algebraic expression in x, we can use a right triangle. Let's consider a right triangle where the angle θ has a sine of x, which means opposite/hypotenuse is x. Since sin⁻¹(x) is θ, by the definition of sine we have a triangle with opposite side of length x and hypotenuse of length 1 (assuming x is within the range [-1,1] since that's the domain of sin⁻¹).
Using the Pythagorean Theorem on this triangle, we can find the length of the adjacent side, which we'll call y. So, we have:
opposite side = x
hypotenuse = 1
adjacent side = √(1 - x²)
The reason the adjacent side is √(1 - x²) is because 1² - x² gives us y², which represents the length of the adjacent side squared.
Now, since the cosine of an angle is the adjacent side over the hypotenuse, cos(θ) is simply y/1 = √(1 - x²). Thus, the algebraic expression for cos(sin⁻¹(x)) is:
√(1 - x²)