Problem 107 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as an algebraic expression in \(u, u>0\). $$\sin \left(\sec ^{-1} \frac{u}{2}\right)$$

Step-by-Step Solution

Answer

\( \sin \left( \sec^{-1} \frac{u}{2} \right) = \sqrt{1 - \frac{4}{u^2}} \)

Step 1: Understand the Inverse Trigonometric Function

Given the expression \( \sin \left( \sec^{-1} \frac{u}{2} \right) \) and we want to find its equivalent algebraic expression, where \( u > 0 \). The \( \sec^{-1} \) function outputs an angle whose secant is \( \frac{u}{2} \).

Step 2: Relate Secant and Cosine

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). If \( \theta = \sec^{-1} \left( \frac{u}{2} \right) \), it means \( \sec \theta = \frac{u}{2} \). Thus, \( \cos \theta = \frac{2}{u} \).

Step 3: Use Pythagorean Identity to Find Sine

We use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = \frac{2}{u} \) into the identity: \( \sin^2 \theta = 1 - \left(\frac{2}{u}\right)^2 \). This simplifies to \( \sin^2 \theta = 1 - \frac{4}{u^2} \).

Step 4: Solve for Sine

Taking the square root of both sides, we consider \( \sin \theta = \sqrt{1 - \frac{4}{u^2}} \). Since \( u > 0 \), \( \sin \theta \) will be positive. So, \( \sin \left( \sec^{-1} \frac{u}{2} \right) = \sqrt{1 - \frac{4}{u^2}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for finding angles whose trigonometric ratios are known. They reverse the process of the standard trigonometric functions. For example, the inverse secant function, denoted as \( \sec^{-1} \), takes a value and returns an angle. This angle is such that its secant value equals the input. So, when we encounter an expression like \( \sec^{-1} \left( \frac{u}{2} \right) \), it means we're looking for an angle, say \( \theta \), where the secant (ratio of the hypotenuse over the adjacent side) is \( \frac{u}{2} \).
Inverse trigonometric functions limit their output to ensure they function properly as inverses. Commonly, with the inverse secant, the angle \( \theta \) lies in the range \([0, \pi] \) (excluding \( \frac{\pi}{2} \)). Understanding these ranges helps avoid ambiguous solutions. So, remember, when working with inverses, think of them as unlocking the angle given a specific trigonometric value.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity involving the sine and cosine functions. It's expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is a direct result of the Pythagorean theorem, hence the name, and is highly useful in simplifying expressions.
In our problem, given that \( \cos \theta = \frac{2}{u} \), we can find \( \sin \theta \) using the Pythagorean identity. To do this, we rearrange the identity to solve for \( \sin^2 \theta \):

• Substitute \( \cos \theta = \frac{2}{u} \) into the identity, we have: \( \sin^2 \theta = 1 - \left( \frac{2}{u} \right)^2 \).
• This results in the simplified form \( \sin^2 \theta = 1 - \frac{4}{u^2} \).

From here, to find \( \sin \theta \), take the positive square root (since \( u > 0 \) implying a positive sine value), giving: \( \sin \theta = \sqrt{1 - \frac{4}{u^2}} \).
The Pythagorean identity shines in these contexts, enabling the transformation of trigonometric expressions into algebraic forms.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the identities are defined. They simplify the process of transforming and solving trigonometric equations.
Several key identities help solve complex problems efficiently. For instance:

• The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
• The reciprocal identity \( \sec \theta = \frac{1}{\cos \theta} \).

In our problem, knowing these identities allowed us to express \( \sin \theta \) in terms of \( u \) when starting from \( \sec^{-1} \left( \frac{u}{2} \right) \). This simplification made use of:

• Reciprocal identity to express cosine from secant: \( \cos \theta = \frac{2}{u} \).
• The Pythagorean identity to solve for sine: \( \sin \theta = \sqrt{1 - \frac{4}{u^2}} \).

Mastering trigonometric identities enhances problem-solving skills highly beneficial in algebra and calculus.