Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{8} $$

Step-by-Step Solution

Answer

\( \frac{2 - \sqrt{2}}{4} \)

Step 1: Apply the Half-Angle Identity

The half-angle identity for sine is given by \( \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2} \). In this problem, \( \theta = \frac{\pi}{4} \) because \( \frac{\pi}{8} = \frac{1}{2} \cdot \frac{\pi}{4} \). So we have: \( \sin^2\left(\frac{\pi}{8}\right) = \frac{1 - \cos\left(\frac{\pi}{4}\right)}{2} \).

Step 2: Calculate \( \cos\left(\frac{\pi}{4}\right) \)

We know that \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) from standard trigonometric values. Substitute this into the identity: \( \frac{1 - \frac{\sqrt{2}}{2}}{2} \).

Step 3: Simplify the Expression

Subtract \( \frac{\sqrt{2}}{2} \) from 1 to find the numerator: \( 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \).

Step 4: Divide by 2

Now, divide the result from Step 3 by 2: \( \frac{2 - \sqrt{2}}{2} \div 2 = \frac{2 - \sqrt{2}}{4} \).

Step 5: Final Answer

Thus, the exact value of \( \sin^2\left(\frac{\pi}{8}\right) \) is \( \frac{2 - \sqrt{2}}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the trigonometric functions. These identities hold true for all angles and can simplify the process of finding unknown values in trigonometric expressions or equations. They include a wide range of equations like Pythagorean identities, sum and difference formulas, and more advanced identities such as the half-angle identities.

• Basic identities include: \( \sin^2(\theta) + \cos^2(\theta) = 1 \) which is the Pythagorean identity.
• Half-angle identities: These allow us to find trigonometric values for an angle divided by 2. For sine, the half-angle identity is \( \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2} \).

Understanding these identities can be extremely helpful when calculating exact trigonometric values, simplifying complex expressions, and solving trigonometric equations.

Sine Function

The sine function, often written as \( \sin(\theta) \), is one of the basic trigonometric functions describing the relationship between the angle and the y-coordinate on the unit circle. It is periodic, with a period of \(2\pi\), meaning it repeats its pattern every \(2\pi\) radians.

• Key properties: The sine function takes values from -1 to 1.
• \( \sin(0) = 0 \), \( \sin\left(\frac{\pi}{2}\right) = 1 \), and \( \sin(\pi) = 0 \)
• Graph behavior: The sine function starts at 0, rises to 1, returns to 0, decreases to -1, and returns to 0 again at \(2\pi\).

The sine function is crucial in various applications, such as modeling waves and oscillations, and is foundational in understanding trigonometric identities, like the half-angle identities used in the original exercise.

Exact Values in Trigonometry

When dealing with trigonometric expressions, sometimes it's necessary to find their exact values rather than approximate decimals. Exact values of trigonometric functions at specific angles often involve well-known simplifications using radical expressions. These come from learned values associated with key angles, such as \(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2},\pi\), among others.

• Common exact values: \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
• Using identities such as the half-angle identities can simplify the process of finding these exact values, as shown in this exercise.
• Benefits of exact values: They provide precise results needed in theoretical mathematics and physics, avoiding the inaccuracies of decimal approximations.

By deploying these tactics, you can determine the exact value of expressions like \( \sin^2\left(\frac{\pi}{8}\right) \) efficiently, ensuring accuracy in mathematical calculations.