Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$

Step-by-Step Solution

Answer

\(\sin^2 \frac{\pi}{6} = \frac{1}{4}\).

Step 1: Recall the Half-Angle Identity for Sine

The half-angle identity for sine is given by: \[\sin^2 \left( \frac{\theta}{2} \right) = \frac{1 - \cos(\theta)}{2}\] This identity will be useful in finding \(\sin^2 \frac{\pi}{6}\).

Step 2: Determine Corresponding Angle

Identify the angle \(\theta\) such that \(\frac{\theta}{2} = \frac{\pi}{6}\). Solving for \(\theta\), multiply both sides by 2:\[\theta = 2 \times \frac{\pi}{6} = \frac{\pi}{3}\] Thus, \(\theta = \frac{\pi}{3}\).

Step 3: Find \(\cos(\theta)\)

Find the cosine of the angle \(\theta = \frac{\pi}{3}\). From the unit circle or trigonometric table, we know:\[\cos \left( \frac{\pi}{3} \right) = \frac{1}{2}\]

Step 4: Substitute Values into the Identity

Substitute \(\cos \left( \frac{\pi}{3} \right)\) into 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.

Step 5: Simplify the Expression

Simplify the expression:\[\frac{1 - \frac{1}{2}}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4}\]Thus, \(\sin^2 \frac{\pi}{6} = \frac{1}{4}\).

Key Concepts

Half-Angle Identity

The half-angle identity is a vital tool in trigonometry that simplifies the process of finding trigonometric values at specific angles. Often, calculations involve angles that are not originally part of the familiar ones found on the unit circle. For these instances, identities like the half-angle identity become particularly useful.

The half-angle identity for the sine function is expressed as: \[\sin^2 \left( \frac{\theta}{2} \right) = \frac{1 - \cos(\theta)}{2}\]
This identity provides a way to determine the square of the sine value for a particular angle that is half of another known angle. By adjusting known trigonometric values, it allows for simpler computation and helps in solving problems where the exact value is needed.

To effectively use this identity, one should:

• Identify the corresponding whole angle, \(\theta\), that, when halved, gives the desired angle.
• Find the cosine of this entire angle, \(\theta\).
• Substitute the cosine value into the formula to find the sine squared of the half-angle.

These steps simplify finding exact trigonometric values when dealing with half-angles.

Sine Function

The sine function is one of the primary trigonometric functions and represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Its values are fundamental in circle and wave studies and are frequently used in mathematical analyses, including solving triangles and modeling periodic phenomena.

For the unit circle, the sine of an angle \(\theta\) is the y-coordinate of the point where the terminal side of the angle intersects the circle. A few noteworthy points about the sine function include:

• Its range is from -1 to 1.
• The sine function is odd, meaning \(\sin(-\theta) = -\sin(\theta)\).
• The function is periodic with a period of \(2\pi\), signifying it repeats every \(2\pi\) radians.

Understanding the sine function's properties and behavior, such as its periodicity and symmetry, helps infer values and solve trigonometric equations efficiently using identities like the half-angle identity.

Trigonometric Values

Trigonometric values are essential for understanding the relationships between angles and sides in triangles. They are also used in more complex applications, such as Fourier transforms and wave equations. Standard trigonometric values are often derived from special angles found frequently in mathematical problems, such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \pi\), and so on.

For example, knowing that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) can make solving trigonometric problems quicker when these values are substituted into identities. Memorization of fundamental trigonometric values helps facilitate calculations where these values or angles are involved.

Key trigonometric ratios for these special angles are often summarized as:

• \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)
• \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
• \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
• \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
• \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
• \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)

Using these ratios, one can seamlessly compute unknown angles, sides, and derive exact values, especially when utilizing trigonometric identities to solve complex problems.