Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The exact value is \(\frac{1}{4}\).

Step 1: Understanding Half-Angle Identity

Recall the half-angle identity for sine: \[\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2}\]Our task is to find \(\sin^2\left(\frac{\pi}{6}\right)\). Therefore, \(\theta = \frac{\pi}{3}\) because \(\frac{\pi}{6}\) is half of \(\frac{\pi}{3}\).

Step 2: Calculating Cosine

Next, we need to find \(\cos\left(\frac{\pi}{3}\right)\), which is a common trigonometric value. From the unit circle, the value is:\[\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\]

Step 3: Applying the Half-Angle Identity

Substitute the cosine value in the half-angle identity:\[\sin^2\left(\frac{\pi}{6}\right) = \frac{1 - \cos\left(\frac{\pi}{3}\right)}{2} = \frac{1 - \frac{1}{2}}{2}\]Simplify the expression:\[\sin^2\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{2} = \frac{1}{4}\]

Step 4: Final Solution

The exact value of \(\sin^2\left(\frac{\pi}{6}\right)\) is \(\frac{1}{4}\). Hence, we have reached our solution.

Key Concepts

Half-Angle Identities

Half-angle identities are a crucial part of trigonometry that help us find values of trigonometric functions for angles that are half of a known angle. These identities stem from angle formulas that simplify the process of solving trigonometry problems.

For example, the sine half-angle identity is given by:

• \[\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2}\]

This formula allows us to calculate the sine of half a given angle if the cosine of the full angle is known. It is particularly beneficial when dealing with common angles found in the unit circle, where cosine values are readily available.

Using half-angle identities can streamline the calculation process, reducing complex trigonometric problems to simpler arithmetic expressions.

Trigonometry Problems

Solving trigonometry problems often involves understanding and utilizing various trigonometric identities and knowledge from the unit circle. These problems can range from simple calculations to complex equations that require a deep understanding of trigonometric properties.

When tackling these problems, it’s critical to have a strategy:

• Identify which trigonometric identities may simplify the problem.
• Assess the known values and relationships, like those from the unit circle.
• Apply identities systematically to solve for the unknown values.

The challenge often lies in recognizing patterns and choosing the correct identity. The half-angle identity, for instance, is an efficient tool when the angle presented is twice the angle that is typically found on the unit circle.

Exact Values in Trigonometry

Finding exact values in trigonometry involves determining known or standard results for trigonometric functions without approximations. These values are essential for precise mathematical calculations.

They usually include values for key angles, such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), and others that yield integer or fractional results when using sine, cosine, or tangent functions.

For example, calculating \(\cos\left(\frac{\pi}{3}\right)\) gives an exact value of \(\frac{1}{2}\).

By using exact values, we ensure mathematical precision, avoiding the rounding errors that are inherent in numerical approximations. This precision is particularly important in mathematics, physics, and engineering applications.

Sine Function Calculations

The sine function is one of the fundamental trigonometric functions, and its calculations often involve transformations and identities to derive specific angle values. A basic calculation method for the sine function is applying identities and known values from the circle.

When given a specific angle, such as \(\frac{\pi}{3}\), we can use this to compute values for various transformed angles using trigonometric identities. For instance, through the half-angle identity, you can find \(\sin^2\left(\frac{\pi}{6}\right)\) by initially determining \(\cos\left(\frac{\pi}{3}\right)\).

The calculation is done as follows:

• Find \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
• Apply the half-angle identity: \(\sin^2\left(\frac{\pi}{6}\right) = \frac{1 - \frac{1}{2}}{2} = \frac{1}{4}\).

Sine calculations like these are simplified and exact when relying on identities, leading to precise and accurate results.