Problem 34 - Precalculus: Functions and Graphs

Question

Express in terms of the cosine function with exponent 1. \(\sin ^{4} \frac{\theta}{2}\)

Step-by-Step Solution

Answer

\(\sin^4 \frac{\theta}{2} = 1 - 2\cos^2 \frac{\theta}{2} + \cos^4 \frac{\theta}{2}\)

Step 1: Use the Pythagorean identity

The Pythagorean identity tells us that for any angle \(\alpha\), \(\sin^2 \alpha + \cos^2 \alpha = 1\). First, express \(\sin^2 \frac{\theta}{2}\) in terms of cosine: \(\sin^2 \frac{\theta}{2} = 1 - \cos^2 \frac{\theta}{2}\).

Step 2: Express \(\sin^4 \frac{\theta}{2}\) using Step 1

We need to express \(\sin^4 \frac{\theta}{2}\) in terms of cosine. Use the identity from Step 1, \(\sin^4 \frac{\theta}{2} = (\sin^2 \frac{\theta}{2})^2 = (1 - \cos^2 \frac{\theta}{2})^2\).

Step 3: Expand the squared term

Expand the expression \((1 - \cos^2 \frac{\theta}{2})^2\). This gives: \[(1 - \cos^2 \frac{\theta}{2})^2 = 1 - 2\cos^2 \frac{\theta}{2} + \cos^4 \frac{\theta}{2}\]

Step 4: Final expression in terms of cosine with exponent 1

The expression \(\sin^4 \frac{\theta}{2}\) in terms of \(\cos \frac{\theta}{2}\) is \(1 - 2\cos^2 \frac{\theta}{2} + \cos^4 \frac{\theta}{2}\).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the square of the sine and cosine of an angle to 1. This identity is expressed as

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

where \(\alpha\) is any angle in a right triangle. This equation comes from the Pythagorean theorem and is vital for solving problems involving trigonometric functions.
By rearranging the identity, you can express sine in terms of cosine, or vice versa. For example, if you solve the identity for \(\sin^2 \alpha\), you get:

• \(\sin^2 \alpha = 1 - \cos^2 \alpha\)

This rearrangement is particularly useful when you need to convert expressions such as \(\sin^4 \left( \frac{\theta}{2} \right)\) into terms of cosine, which can simplify the algebraic manipulations in the problem.
Understanding how to utilize the Pythagorean identity is key for transitioning between sine and cosine in various problems.

sine and cosine relationship

Sine and cosine have a unique relationship in trigonometry, where they are both projections of the unit circle onto the y-axis and x-axis, respectively. This relationship not only underlies the Pythagorean identity but also allows for a direct transformation between the two functions.

For example:

• If you know \(\cos \alpha\), you can find \(\sin \alpha\) using the identity \(\sin^2 \alpha = 1 - \cos^2 \alpha\).
• Similarly, \(\cos^2 \alpha\) can be rewritten as \(1 - \sin^2 \alpha\).

In the context of the original exercise, utilizing the sine to cosine conversion is essential for expressing powers of sine in terms of cosine. When you work through expressions like \(\sin^4 \left( \frac{\theta}{2} \right)\), initially reducing \(\sin^2 \left( \frac{\theta}{2} \right)\) to \(1 - \cos^2 \left( \frac{\theta}{2} \right)\) is a powerful step in simplifying complex trigonometric expressions using fewer function types.

algebraic manipulation

Algebraic manipulation involves using algebraic methods to transform expressions into a new, often simpler form. This skill is crucial in many areas of mathematics, including trigonometry.

In dealing with trigonometric identities, algebraic manipulation allows you to expand, simplify, or factor expressions to reveal insights or simplify calculations. For example, expanding \((1 - \cos^2 \frac{\theta}{2})^2\) for solving \(\sin^4 \left( \frac{\theta}{2} \right)\) in the original exercise employs:

• The distributive property: \((a - b)^2 = a^2 - 2ab + b^2\)
• Substituting terms: transforming \(1 - \cos^2 \left( \frac{\theta}{2} \right)\) to \(\sin^2 \left( \frac{\theta}{2} \right)\)

Expanding and simplifying allows breaking down complex expressions such as \(\sin^4 \left( \frac{\theta}{2} \right) = 1 - 2\cos^2 \frac{\theta}{2} + \cos^4 \frac{\theta}{2}\), which helps to present the function only in terms of cosine. Mastery of these concepts aids significantly in manipulating and solving trigonometric identities efficiently. This practice not only simplifies calculations but is also essential for deeper mathematical problem solving.