Problem 6 - Precalculus with Calculus Previews

Question

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos \frac{11 \pi}{12} $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Identify Angles for Sum or Difference Formulas

To use the sum or difference formula for cosine, we first need to express \( \frac{11\pi}{12} \) as the sum or difference of two special angles whose cosine values we know. \( \frac{11\pi}{12} \) can be rewritten as the sum \( \frac{11\pi}{12} = \frac{3\pi}{4} + \frac{\pi}{6} \).

Step 2: Apply the Cosine Sum Formula

The cosine of the sum of two angles is given by the formula \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Substitute \( a = \frac{3\pi}{4} \) and \( b = \frac{\pi}{6} \) into the formula.

Step 3: Find Values for Individual Trigonometric Functions

Determine the trigonometric values of \( \cos \frac{3\pi}{4} \), \( \cos \frac{\pi}{6} \), \( \sin \frac{3\pi}{4} \), and \( \sin \frac{\pi}{6} \).\[ \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2} \].

Step 4: Substitute Values into Formula

Substitute these values into the sum formula: \[ \cos \left( \frac{3\pi}{4} + \frac{\pi}{6} \right) = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \].

Step 5: Calculate the Expression

Compute the above expression: \[ -\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = -\frac{\sqrt{6}}{4} \] and \[ \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{2}}{4} \]. Therefore, combining these gives: \[ -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \].

Step 6: Write the Conclusion

The exact value of \( \cos \frac{11\pi}{12} \) using the sum formula and known angle values is \( -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Sum Formula

The sum formula in trigonometry is incredibly useful when working with angles that are not directly given as special angles. When we want to find the cosine of a sum of two angles, we use the formula:

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

This formula helps us break down complex angles into simpler ones that we can evaluate more readily using known values. For example, to find \( \cos \frac{11\pi}{12} \), we express it as the sum of two special angles. These special angles are ones with exact cosine and sine values that can be easily recalled or derived.
Using the formula involves a step-by-step process:

• Identify the angles \( a \) and \( b \) that make up \( a+b=\frac{11\pi}{12} \).
• Apply the sum formula by inserting the cosine and sine values of the special angles \( a \) and \( b \).
• Compute the resulting expression to find the exact trigonometric value needed.

Difference Formula

While the sum formula is applicable for the given exercise, understanding difference formulas also adds to the repertoire of solving various problems. The cosine of the difference of two angles can be found using the formula:

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

Knowing both sum and difference formulas enables us to work with any configuration of angles. Even if one approach is favored in a particular context, being familiar with both increases mathematical flexibility. Whenever you're handling exercises that involve trigonometric functions, these formulas allow you to manipulate angles into forms that are much easier to manage, leveraging the exact values from special angles.

Exact Values

Exact values are essential in trigonometry, providing the precise trigonometric values for certain notable angles. These values are derived from unique angles commonly found in geometry problems, such as \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \).
For instance, in the step-by-step solution:

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

These values come from basic triangles and the unit circle. Memorizing and understanding these values are crucial as they often serve as building blocks for more complex trigonometric evaluations.

Special Angles

Special angles are specific angles frequently used in trigonometry that have known exact trigonometric values. These include \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and their combinations in other quadrants.
Understanding these angles and their corresponding trigonometric values can greatly simplify calculations:

• Each special angle has well-defined cosine, sine, and tangent values that are easy to recall from memory.
• They help in forming sums and differences that can break down more complex angles into familiar terms.

In the example, \( \frac{11\pi}{12} \) was expressed as a sum of \( \frac{3\pi}{4} \) and \( \frac{\pi}{6} \), both of which are special angles. Recognizing and working with these angles allows us to solve trigonometric problems without a calculator efficiently.