Problem 103 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\( \sin (\arccos u) = \sqrt{1-u^2} \)

Step 1: Understand Inverse Trigonometric Functions

Recognize that \( \arccos u \) is the angle whose cosine is \( u \). So we let \( \theta = \arccos u \), which implies that \( \cos \theta = u \).

Step 2: Use the Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Since \( \cos \theta = u \), we find \( \sin^2 \theta = 1 - u^2 \).

Step 3: Solve for Sine

To find \( \sin \theta \), take the square root of both sides: \( \sin \theta = \sqrt{1 - u^2} \). Because \( u > 0 \) and \( \theta \) is in the range \([0, \pi]\), which ensures \( \sin \theta \geq 0 \), we can write \( \sin (\arccos u) = \sqrt{1 - u^2} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most important trigonometric identities, fundamental in simplifying trigonometric expressions and solving equations. It states that for any angle \( \theta \), the sum of the square of sine and cosine is always 1. In mathematical form, this identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity stems from the Pythagorean Theorem and holds true for all real values of \( \theta \). When dealing with inverse trigonometric functions, such as \( \arccos u \), this identity becomes very useful. Since \( \theta = \arccos u \) implies \( \cos \theta = u \), we use the identity to express \( \sin^2 \theta \) in terms of \( u \).

By substituting \( \cos \theta = u \) into the identity, we get \( \sin^2 \theta = 1 - u^2 \). This step allows us to determine \( \sin \theta \) by taking the square root, thus transforming a trigonometric function into an algebraic expression. This transformation is essential when dealing with inverse trigonometric functions that require rewriting in terms of algebraic expressions.

Algebraic Expressions

Writing trigonometric expressions as algebraic expressions is a useful skill, making complex functions more manageable. In our given problem, we start with \( \sin(\arccos u) \). Here, \( \arccos u \) represents an angle whose cosine equals \( u \).

By understanding that \( \theta = \arccos u \) means \( \cos \theta = u \), we can use the Pythagorean Identity to transform \( \sin^2 \theta = 1 - u^2 \). Solving for \( \sin \theta \), we find:

• \( \sin \theta = \sqrt{1 - u^2} \)

This result implies that the original expression can be simplified to an algebraic form, \( \sqrt{1 - u^2} \), which is much easier to interpret and calculate. Converting trigonometric expressions into algebraic ones often helps in further calculations, ensuring clarity and simplicity.

Trigonometric Identities

Trigonometric identities are equations that relate trigonometric functions to each other. These identities are valid for all angle measures and serve as powerful tools in algebraic transformations. Some of the most common identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle Sum and Difference Identities
• Double Angle Identities

In our exercise, the Pythagorean Identity is particularly useful. It allows us to express one trigonometric function, \( \sin(\theta) \), in terms of another, \( \cos(\theta) \), thereby providing an algebraic solution.

By understanding and applying these identities, you can simplify complex expressions, making them solvable even if they initially appear intricate. Remember, the facility to interchange between trigonometric and algebraic expressions through identities is a critical skill in many areas of mathematics, such as calculus and physics.