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

Verified step by step guidance

1. Recognize that \( \arccos(u) \) is the angle whose cosine is \( u \). Let's call this angle \( \theta \), so \( \cos(\theta) = u \).
2. Use the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
3. Substitute \( \cos(\theta) = u \) into the identity: \( \sin^2(\theta) + u^2 = 1 \).
4. Solve for \( \sin^2(\theta) \): \( \sin^2(\theta) = 1 - u^2 \).
5. Take the square root to find \( \sin(\theta) \): \( \sin(\theta) = \sqrt{1 - u^2} \), since \( u > 0 \) implies \( \sin(\theta) \geq 0 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find angles when given a trigonometric ratio. For example, if u = cos(θ), then θ = arccos(u). Understanding how to interpret these functions is crucial for converting expressions involving them into algebraic forms.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is essential when working with trigonometric functions, as it allows us to express sine in terms of cosine and vice versa, facilitating the conversion of expressions like sin(arccos(u)) into algebraic forms.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. For instance, in a right triangle, sin(θ) is defined as the ratio of the opposite side to the hypotenuse. Recognizing these ratios helps in transforming trigonometric expressions into algebraic ones, particularly when dealing with angles derived from inverse functions.