Problem 29 - Precalculus Mathematics for Calculus

Question

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(2 \sin 18^{\circ} \cos 18^{\circ}\) (b) \(2 \sin 3 \theta \cos 3 \theta\)

Step-by-Step Solution

Answer

(a) Simplifies to \( \sin 36^{\circ} \); (b) simplifies to \( \sin 6\theta \).

Step 1: Identify the Formula Needed

We need to simplify both expressions, which are in the form of products of sine and cosine. The appropriate formula for this situation is the **Double-Angle identity** for sine, which states \( \sin 2\theta = 2 \sin \theta \cos \theta \).

Step 2: Apply the Double-Angle Formula to Part (a)

In the expression \( 2 \sin 18^{\circ} \cos 18^{\circ} \), set \( \theta = 18^{\circ} \). Applying the formula \( \sin 2\theta = 2 \sin \theta \cos \theta \), we identify that \( 2 \sin 18^{\circ} \cos 18^{\circ} = \sin(2 \times 18^{\circ}) = \sin 36^{\circ} \).

Step 3: Apply the Double-Angle Formula to Part (b)

For the expression \( 2 \sin 3\theta \cos 3\theta \), set \( \theta = 3\theta \). Again using the formula \( \sin 2\theta = 2 \sin \theta \cos \theta \), we find that \( 2 \sin 3\theta \cos 3\theta = \sin(2 \times 3\theta) = \sin(6\theta) \).

Key Concepts

Double-Angle Formula

The Double-Angle Formula is a crucial tool in trigonometry. It helps simplify expressions involving trigonometric functions. One of its most common forms specifies the sine of a double angle.

• Formula: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• Purpose: Used to reduce products of sine and cosine into single trigonometric functions.
• Applications: Simplifies complex trigonometric equations and expressions, making them easier to manage.

To use this formula, identify the angle \( \theta \) and apply the formula to find the sine of twice that angle. For example, transforming \( 2 \sin 18^{\circ} \cos 18^{\circ} \) to \( \sin 36^{\circ} \) involves straightforward manipulation using the Double-Angle Formula.

Half-Angle Formula

The Half-Angle Formulas allow the computation of trigonometric functions at half of a given angle. While not directly used in the original exercise, they are vital in trigonometry.Consider they can break down complex functions into more recognizable forms:

• Formulas for Sine and Cosine: \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \) and \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
• Applications: Essential for solving integrals and during harmonic analysis.

When needing to solve for a trigonometric function at half its usual angle, these formulas become particularly useful, simplifying equations involving angles.

Sine Function

The sine function, represented as \( \sin \theta \), is one of the fundamental components of trigonometry.

• Definition: It measures the y-coordinate in the unit circle of an angle \( \theta \). Often visualized as a wave, the sine function oscillates between -1 and 1.
• Properties: It is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.
• Usefulness: The sine function is used to model cyclical phenomena like sound waves and tides.

In trigonometry, the sine function frequently operates with other functions, like cosine, especially in identities like the Double-Angle Formula. Learning how these functions interplay is key to mastering trigonometric equations and applications.