Problem 28 - Precalculus Mathematics for Calculus

Question

Use an appropriate Half-Angle Formula to find the exact value of the expression. $$\sin \frac{11 \pi}{12}$$

Step-by-Step Solution

Answer

The exact value of \( \sin \frac{11\pi}{12} \) is \( \frac{\sqrt{2 - \sqrt{3}}}{2} \).

Step 1: Identify the appropriate angle

Recognize that the angle \( \frac{11\pi}{12} \) is a half-angle. Noticing that \( \frac{11\pi}{12} = \frac{\pi}{6} + \frac{\pi}{4} \) can help us rewrite it to use a half-angle identity.

Step 2: Rewrite using half-angle relation

The angle \( \frac{11\pi}{12} \) can be expressed in terms of its half-angle representation: \( \frac{11\pi}{12} = \pi - \frac{\pi}{12} \). Thus, \( \frac{\pi}{12} \) is the half-angle we need to focus on for finding \( \sin \frac{11\pi}{12} \).

Step 3: Recall the half-angle sine formula

The sine half-angle formula is \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \). For our expression, \( \theta = \frac{\pi}{6} \), so \( \sin \frac{\pi}{12} = \sqrt{\frac{1 - \cos \frac{\pi}{6}}{2}} \).

Step 4: Calculate \( \cos \frac{\pi}{6} \)

Recall that the cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \). This value is needed to substitute into the half-angle formula.

Step 5: Substitute and compute

Now substitute \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) into the half-angle formula: \( \sin \frac{\pi}{12} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \). Simplify this to get \( \sin \frac{\pi}{12} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \).

Step 6: Reflect consideration for \( \pi - \frac{\pi}{12} \)

Since \( \sin(\pi - x) = \sin x \), we apply this to find that \( \sin \frac{11\pi}{12} = \sin \frac{\pi}{12} \). Therefore, \( \sin \frac{11\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of a triangle in terms of trigonometric functions.
These identities help us solve complex problems in trigonometry by providing shortcuts and simplifying expressions.

• Some common trigonometric identities include the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle sum and difference identities: For example, \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)
• And the double angle identities: Like \( \sin(2\theta) = 2\sin \theta \cos \theta \)

These identities are incredibly useful as they allow us to transform trigonometric expressions into forms that are more easily solvable. The half-angle identities, for example, which are used in the given exercise, allow us to find trigonometric values at angles that are not easily accessible otherwise.

Angle Measurement

Angles can be measured in degrees or radians, and each unit has its own advantages. Radians are often preferred in higher mathematics because they relate more naturally to other mathematical concepts.
In the context of radians:

• 1 full revolution is measured as \( 2\pi \) radians
• A right angle measures \( \frac{\pi}{2} \) radians
• A straight angle is \( \pi \) radians

An important thing to remember is that when an angle like \( \frac{11\pi}{12} \) appears, the task is often to recognize how it can be expressed in terms of basic angles for which we know standard trigonometric values. In exercises, often you'll break down complex angles to simpler fractions of \( \pi \) to use known identities like the half-angle formulas.

Sine Function

The sine function is a fundamental trigonometric function defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.
It is one of the primary functions used to describe wave forms and oscillations among other phenomena.

• Mathematically, the sine of an angle \( \theta \) is given by: \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• Its graph is a smooth, continuous wave that repeats every \( 2\pi \) radians

The sine function has a range from -1 to 1, and its important symmetry property includes \( \sin(\pi - x) = \sin x \). This characteristic is useful as it allows us to simplify expressions and solve equations.
In this exercise, by recognizing the symmetrical properties, one finds \( \sin \frac{11\pi}{12} = \sin \frac{\pi}{12} \).

Exact Values in Trigonometry

While calculators are great for approximations, finding exact values is key in trigonometry, especially when precise solutions are needed. These values come from specific angles where trigonometric functions have well-known "nice" values.

• Some of these angles include:\( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \)
• Associated exact sine values include:\( 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1 \)

By using identities such as half-angle formulas, you can evaluate trigonometric functions at these angles or their related fractions.
The half-angle formula for sine, \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \), provides a way to find exact sine values for angles like \( \frac{\pi}{12} = \frac{1}{2} \times \frac{\pi}{6} \). This helps derive an exact value \( \sin \frac{11\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2} \). These values not only help in academia but also have practical applications in engineering and physics.