Problem 13 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \cos \frac{11 \pi}{12} $$

Step-by-Step Solution

Answer

\( \cos \frac{11\pi}{12} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Identify Angle Pair

To use a cosine addition or subtraction formula, we need to express \( \frac{11\pi}{12} \) as a sum or difference of angles we know the cosine values for. Notice \( \frac{11\pi}{12} = \pi - \frac{\pi}{12} \).

Step 2: Apply Cosine Subtraction Formula

The cosine subtraction formula is \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Here \( a = \pi \) and \( b = \frac{\pi}{12} \), so \( \cos \pi = -1 \) and \( \sin \pi = 0 \).

Step 3: Find Values for Components

We also need \( \cos \frac{\pi}{12} \) and \( \sin \frac{\pi}{12} \). Let's use known angle values: \( \cos \frac{\pi}{12} = \cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \) and \( \sin \frac{\pi}{12} = \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 4: Substitute Values Into Formula

Substitute these values into the formula: \[ \cos \frac{11\pi}{12} = (-1) \cdot \frac{\sqrt{6} + \sqrt{2}}{4} + 0 \cdot \frac{\sqrt{6} - \sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4}. \]

Step 5: Simplify the Expression

The result simplifies to \( \cos \frac{11\pi}{12} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Cosine Addition Formula

When it comes to trigonometry, the cosine addition formula is a useful tool for finding the cosine of the sum or difference of two angles. This formula can help simplify calculations and obtain exact values, which are essential for precise results in mathematics. The cosine subtraction formula is written as:\[\cos(a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b\]Here's how it works:

• Subtract Angles: It involves two angles, \(a\) and \(b\), where the expression is like a difference i.e., \(a - b\).
• Trigonometric Products: The formula breaks down into products of the cosine and sine of these angles.

In the given exercise, \(\frac{11\pi}{12}\) is rewritten as a difference, specifically \(\pi - \frac{\pi}{12}\). Using the cosine subtraction formula, this becomes \(\cos \pi \cdot \cos \frac{\pi}{12} + \sin \pi \cdot \sin \frac{\pi}{12}\). With \(\cos \pi = -1\) and \(\sin \pi = 0\), we further simplify the expression, showcasing the power and simplicity of using the cosine addition formula.

Exact Trigonometric Values

Exact trigonometric values help us calculate angles without approximating with decimals, which can lead to inaccuracies. These values are well-known for certain angles and are derived from specific trigonometric identities.Common angles, like 30°, 45°, and 60°, or their radian equivalents \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), have well-established exact trigonometric values. For instance, \(\cos 45° = \frac{\sqrt{2}}{2}\).In the exercise at hand, we encounter \(\cos 15°\) and \(\sin 15°\), which aren't as common, but they have exact values too:

• \(\cos 15°\) (or \(\cos \frac{\pi}{12}\)): \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
• \(\sin 15°\) (or \(\sin \frac{\pi}{12}\)): \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

These precise values are particularly helpful for calculations in the cosine subtraction formula, ensuring the final result is exact.

Radians and Degrees

Understanding the distinction between radians and degrees is key in trigonometry. These are the two units of measuring angles, with radians often used in higher mathematics for more straightforward calculus operations.

• Degrees: A complete circle is 360°, making it intuitive and widely used in various fields.
• Radians: A complete circle is \(2\pi\) radians. Radians originate from the relationship between the radius and the circumference of a circle.

To convert from degrees to radians:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]Conversely, for converting radians to degrees:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]In the given problem, \(\frac{11\pi}{12}\) is already in radians. Converting between these units simplifies the process of identifying and using the exact trigonometric values and applying formulas effectively.