Write each trigonometric expression as an algebraic expression in u, for u > 0.
cos (arcsin u)

Verified step by step guidance

1. Recognize that the expression is \( \cos(\arcsin u) \). Here, \( \arcsin u \) represents an angle \( \theta \) such that \( \sin \theta = u \).
2. Set \( \theta = \arcsin u \), so by definition, \( \sin \theta = u \). Since \( u > 0 \), \( \theta \) is in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) where cosine is non-negative.
3. Use the Pythagorean identity for sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = u \) to get \( u^2 + \cos^2 \theta = 1 \).
4. Solve for \( \cos \theta \): \( \cos \theta = \sqrt{1 - u^2} \). Since \( \theta \) is in the first or fourth quadrant (due to the range of arcsin), cosine is positive, so take the positive root.
5. Therefore, \( \cos(\arcsin u) = \sqrt{1 - u^2} \), which is the algebraic expression in terms of \( u \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin, return the angle whose trigonometric ratio equals a given value. For example, arcsin(u) gives an angle θ such that sin(θ) = u. Understanding this allows us to rewrite expressions involving inverse functions in terms of angles.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This relationship helps express one trigonometric function in terms of another, such as finding cos(θ) when sin(θ) is known, which is essential for rewriting cos(arcsin u) algebraically.

Domain and Range Considerations

When dealing with inverse trigonometric functions, it's important to consider the domain and range to determine the correct sign of the resulting expression. Since u > 0 and arcsin(u) lies in [-π/2, π/2], cos(arcsin u) will be non-negative, guiding the choice of the positive root in the algebraic expression.