Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f(\theta)=\sin 2 \theta, 0<\theta<\frac{\pi}{4}\)

Step-by-Step Solution

Answer

Function is increasing from 0 to 1 for \(0 < \theta < \frac{\pi}{4}\).

Step 1: Identify the Function

Given the function is \( f(\theta) = \sin 2\theta \). The domain constraint is \( 0 < \theta < \frac{\pi}{4} \). We are dealing with a trigonometric function.

Step 2: Apply the Double Angle Formula

The function \( f(\theta) = \sin 2\theta \) involves a double angle. Recall the double angle identity for sine: \( \sin 2\theta = 2\sin\theta\cos\theta \). This identity will help us express the function in terms of \( \theta \).

Step 3: Verify the Interval

Given \( 0 < \theta < \frac{\pi}{4} \), calculate the interval for \( 2\theta \). Since \( \theta \) is multiplied by 2, the interval changes to \( 0 < 2\theta < \frac{\pi}{2} \). Ensure any further steps respect this range.

Step 4: Analyze the Function

We now know \( f(\theta) = \sin 2\theta \) evaluated in the domain \( 0 < \theta < \frac{\pi}{4} \) will increase as \( \theta \) increases, given that \( \sin \) increases between \( 0 \) and \( \frac{\pi}{2} \). Therefore, \( \sin 2\theta \) will also increase from \( \sin 0 = 0 \) to \( \sin \frac{\pi}{2} = 1 \).

Key Concepts

Double Angle Formula

The double angle formula is a crucial concept in trigonometry that allows us to express trigonometric functions involving double angles in terms of single angles. A common example is the double angle formula for sine, which is given by:

• \( \sin 2\theta = 2\sin\theta\cos\theta \).

This formula is particularly useful for simplifying expressions or solving equations that involve the sine of a double angle. To understand it better, consider the derivation:

• Start with the identities \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
• By letting \( A = B = \theta \), we derive \( \sin(\theta + \theta) = \sin\theta \cos\theta + \cos\theta \sin\theta \), which simplifies to \( 2\sin\theta\cos\theta \).

The double angle formula is not limited to sine alone; cosine and tangent also have double angle identities, such as \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) or \( 1 - 2\sin^2\theta \), and \( \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta} \). These formulas are instrumental in calculating and simplifying trigonometric functions.

Sine Function

The sine function is one of the fundamental trigonometric functions, represented as \( \sin\theta \). It describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. As a wave function, it oscillates between -1 and 1, making it cyclic every \( 2\pi \) radians.

Some important properties of the sine function include:

• **Periodicity:** The sine function has a period of \( 2\pi \), repeating its values after every complete cycle.
• **Amplitude:** The peak value of the sine function is 1, which means it oscillates between -1 and 1.
• **Symmetry:** Sine is an odd function, which means \( \sin(-\theta) = -\sin\theta \).

The sine function is vital in modeling natural phenomena such as sound waves, light waves, and alternating currents. Understanding its behavior across different intervals helps in analyzing its applications in real-world scenarios, especially when combined with identities like the double angle formula.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for specified angles. They are essential tools in simplifying and solving trigonometric equations.

Some key trigonometric identities include:

• **Pythagorean Identities:** \( \sin^2\theta + \cos^2\theta = 1 \).
• **Reciprocal Identities:** \( \csc\theta = \frac{1}{\sin\theta} \), \( \sec\theta = \frac{1}{\cos\theta} \), and \( \cot\theta = \frac{1}{\tan\theta} \).
• **Angle Sum and Difference Identities:** \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \), etc.

These identities not only simplify calculations but also facilitate the transformation of complex trigonometric expressions into manageable forms. For example, the double angle formula is derived from the angle sum identities. Trigonometric identities are widely used in physics for wave analysis, electrical engineering for analyzing currents and voltages, and even in computer graphics for rendering objects effectively.