Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \left(\frac{13 \pi}{12}\right)$$

Step-by-Step Solution

Answer

\( \sin \left(\frac{13\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1: Identify an Appropriate Identity

To find the exact value of \( \sin \left(\frac{13\pi}{12}\right) \), use the sum identity for sine because \( \frac{13\pi}{12} \) is not a special angle itself. The sum identity is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Express the Angle as a Sum of Special Angles

Since \( \frac{13\pi}{12} = \pi + \frac{\pi}{12} \), we can express it as a sum of angles where \( a = \pi \) and \( b = \frac{\pi}{12} \). Now, let’s express \( \pi \) in radians: \( \pi = 12\frac{\pi}{12} \). Thus, \( \frac{13\pi}{12} = \pi + \frac{\pi}{12} \).

Step 3: Calculate Using the Identity

Apply the identity \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) where \( a = \pi \) and \( b = \frac{\pi}{12} \). Since \( \sin \pi = 0 \) and \( \cos \pi = -1 \), the expression becomes \( 0 \cdot \cos \left(\frac{\pi}{12}\right) + (-1) \cdot \sin \left(\frac{\pi}{12}\right) = -\sin \left(\frac{\pi}{12}\right) \).

Step 4: Find the Value of Special Angles

Use a known identity or derived values to compute \( \sin \left(\frac{\pi}{12}\right) \). We can express \( \frac{\pi}{12} \) as \( \frac{\pi}{6} - \frac{\pi}{4} \). Thus \( \sin \left(\frac{\pi}{12}\right) = \sin(\frac{\pi}{6} - \frac{\pi}{4}) \).

Step 5: Use the Difference Formula for Sine

Using the identity \( \sin(a - b) = \sin a \cos b - \cos a \sin b \), where \( a = \frac{\pi}{6} \) and \( b = \frac{\pi}{4} \), apply the values: \( \sin \frac{\pi}{6} = \frac{1}{2} \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), and \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). This gives \( \sin \left(\frac{\pi}{12}\right) = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \).

Step 6: Simplify the Expression

Simplifying the expression: \( \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \) and \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} \), thus \( \sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \).

Step 7: Calculate the Final Value

Therefore, \( \sin \left(\frac{13\pi}{12}\right) = -\sin \left(\frac{\pi}{12}\right) = -\left(\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\right) = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Key Concepts

Sine Function

The sine function is a fundamental building block in trigonometry, often represented as \( \sin \theta \). It varies between -1 and 1 and describes the vertical component of an angle in a right triangle relative to the hypotenuse. Furthermore, in the unit circle, the sine of an angle is the y-coordinate of the corresponding point on the circle. This function is periodic and repeats every \(2\pi\) radians, meaning:

• If \( \theta \) is an angle, then \( \sin(\theta + 2\pi k) = \sin(\theta) \) for any integer \( k \).
• The sine function is odd, implying \( \sin(-\theta) = -\sin(\theta) \).

Understanding these basic properties assists in solving trigonometric equations and finding values of sine for various angles.

Angle Sum Formula

The angle sum formula is a vital trigonometric identity essential for calculating the sine or cosine of an angle expressed as a sum or difference. For sine, the formula is given by:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)

This identity allows us to break down complex angles into simpler, known ones. For example, using special angles such as \( \pi/3 \), \( \pi/4 \), and \( \pi/6 \), we can derive sine values of non-standard angles. This is particularly useful in exercises like finding \( \sin \left(\frac{13\pi}{12}\right) \), which isn't straightforward without such identities.

Exact Value Calculation

Finding the exact value of trigonometric expressions is a frequent requirement in math, particularly when calculator use is restricted. The goal here is to express an angle in terms of known special angles, using identities like the angle sum or difference formulas.

The process typically involves:

• Expressing the angle as a combination of special angles.
• Applying trigonometric identities, such as the sine and cosine addition and subtraction formulas, to simplify the expression.
• Simplifying further by calculating the values based on well-known angles, such as \( \sin(\pi/6) = \frac{1}{2} \) or \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \).

For example, for \( \sin \left(\frac{13\pi}{12}\right) \), the exact calculation involves decomposing into \( \pi + \frac{\pi}{12}\) then using the subtraction formula to find the precise value of the sine expression.