Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{8} $$

Step-by-Step Solution

Answer

The exact value of \( \sin^2 \frac{\pi}{8} \) is \( \frac{2 - \sqrt{2}}{4} \).

Step 1: Identify the Half-Angle Identity

To find \( \sin^2 \frac{\pi}{8} \), we start by identifying the relevant half-angle identity. Remember, the half-angle identity for sine is \( \sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}\).

Step 2: Set Up the Expression

In this context, \( \theta = \frac{\pi}{4} \) since \( \frac{\pi}{8} \) is half of \( \frac{\pi}{4} \). Substitute \( \theta = \frac{\pi}{4} \) into the half-angle identity: \( \sin^2 \frac{\pi}{8} = \frac{1 - \cos \frac{\pi}{4}}{2}\).

Step 3: Calculate \(\cos \frac{\pi}{4}\)

Recall that \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Substitute this value into the expression: \( \sin^2 \frac{\pi}{8} = \frac{1 - \frac{\sqrt{2}}{2}}{2}\).

Step 4: Simplify the Expression

Simplify the expression: \( \sin^2 \frac{\pi}{8} = \frac{1 - \frac{\sqrt{2}}{2}}{2} = \frac{2/2 - \sqrt{2}/2}{2} = \frac{2 - \sqrt{2}}{4}\).

Step 5: Conclusion: Final Simplified Result

The final simplified result for \( \sin^2 \frac{\pi}{8} \) is \( \frac{2 - \sqrt{2}}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. They are fundamental tools for simplifying complex expressions and solving trigonometric equations. One of the most useful categories of these identities is the half-angle identities, which help us find the sine, cosine, and tangent of half an angle using the known values of these functions at the full angle. For example, the half-angle identity for sine is: - \( \sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2} \) Using this identity, we can simplify and calculate expressions like \( \sin^2 \frac{\pi}{8} \) by knowing the value of \( \cos \theta \). The use of these identities simplifies complex angle calculations into more manageable components, making them a powerful ally in trigonometry.

Sine Function

The sine function is one of the primary trigonometric functions, represented as \( \sin \theta \). It measures the y-coordinate of the unit circle corresponding to an angle \( \theta \). As part of the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \), it fundamentally connects to the cosine function.The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. It is essential in modeling wave patterns and oscillatory phenomena across various scientific fields. In problems involving half-angles, such as \( \sin^2 \frac{\pi}{8} \), we use identities to simplify expressions and calculate values by breaking them into more manageable parts. This is especially useful when dealing with precise measurements or where calculators may not provide exact values.

Angle Simplification

Angle simplification is a crucial concept in trigonometry, allowing us to understand and solve problems involving angles that may not initially appear straightforward. Simplification often involves using identities to express complex angles, such as \( \frac{\pi}{8} \), in terms of more familiar ones like \( \frac{\pi}{4} \) or \( \frac{\pi}{2} \). In the given problem, \( \frac{\pi}{8} \) is expressed as half of \( \frac{\pi}{4} \). This conversion enables us to apply the half-angle identity for sine: - \( \sin^2 \frac{\pi}{8} = \frac{1 - \cos \frac{\pi}{4}}{2} \) Through this simplification process, we achieve exact values that would otherwise be difficult to compute. This technique is particularly useful in trigonometry, where precise calculation is often necessary for accurate analysis.