Problem 32 - Algebra and Trigonometry

Question

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}} & {\text { (b) } 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \cos \theta \); (b) \( \sin \theta \).

Step 1: Identify the Formula for Part (a)

For the expression \( \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} \), we recognize this as a difference of squares. Specifically, it is analogous to \( \cos^2 x - \sin^2 x \), which is the cosine double angle formula: \( \cos 2x = \cos^2 x - \sin^2 x \). Therefore, \( \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} = \cos \theta \).

Step 2: Solution for Part (a)

Using the identified double-angle formula, simplify the expression: \[ \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} = \cos \theta \].

Step 3: Identify the Formula for Part (b)

The expression \( 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \) resembles the sine double angle formula: \( \sin 2x = 2 \sin x \cos x \). Thus, \( 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \sin \theta \).

Step 4: Solution for Part (b)

Apply the sine double-angle formula to simplify: \[ 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \sin \theta \].

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with angles and circular functions. These identities express relationships between trigonometric functions, helping simplify and solve complex problems.

Some important identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Sum and Difference Formulas: These are used to find the sine, cosine or tangent of angles summed or subtracted.
• Double-Angle and Half-Angle Formulas: Allow us to express trigonometric functions of multiple angles or fractions of angles in terms of functions of a single angle.

By understanding these identities, one can transform seemingly complicated expressions into simpler forms. This is particularly beneficial in calculus, physics, and engineering where trigonometry is frequently applied.

Cosine Double-Angle Formula

The cosine double-angle formula is a specific trigonometric identity that helps simplify expressions involving the cosine of double angles. The formula is given by:

• \( \cos 2x = \cos^2 x - \sin^2 x \)
• Additionally, \( \cos 2x \) can also be expressed in alternate forms: \( 2\cos^2 x - 1 \) or \( 1 - 2\sin^2 x \).

The primary use of this formula is to rewrite expressions involving squared trigonometric functions into a single cosine function of a double angle.

In practical problems, like the given exercise part (a), this formula allows us to simplify \( \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} \) to \( \cos \theta \), making calculations easier and the expression more manageable.

Sine Double-Angle Formula

The sine double-angle formula provides a way to simplify expressions involving the sine of a doubled angle. The formula is expressed as:

• \( \sin 2x = 2 \sin x \cos x \)

This identity is particularly useful when transforming a product of sine and cosine into a single sine function of a doubled angle.

In exercises like part (b) of the original problem, the sine double-angle formula allows us to simplify \( 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \) to \( \sin \theta \). This simplification helps reduce complexity, facilitating easier manipulation of trigonometric expressions in calculations or proofs.

Half-Angle Formulas

The half-angle formulas are trigonometric identities that express trigonometric functions of half an angle in terms of the square roots of expressions involving that angle. Given their nature, these formulas are particularly helpful in solving trigonometric equations or simplifying expressions.

• \( \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} \)
• \( \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \)
• \( \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \frac{\sin x}{1 + \cos x} \)

These formulas are powerful tools for angle reduction, especially when you need to solve equations that involve specific angles or simplify integrals and other expressions. Always take note of the sign, which depends on the quadrant in which the angle lies, ensuring that the results are accurate.