Problem 9 - Contemporary Precalculus

Question

$$\text {Find the exact value.}$$ $$\cos \frac{11 \pi}{12}$$

Step-by-Step Solution

Answer

Answer: The exact value of $$\cos\left(\frac{11 \pi}{12}\right)$$ is $$-\frac{\sqrt{2}+\sqrt{6}}{4}$$.

Step 1: Rewrite the given angle using standard angles

We can rewrite $$\frac{11 \pi}{12}$$ as the difference of two known standard angles, $$\frac{\pi}{3}$$ and $$\frac{\pi}{4}$$: $$\frac{11 \pi}{12} = \frac{4 \pi}{12} + \frac{7 \pi}{12} = \frac{\pi}{3} + \frac{7 \pi}{12} = \left(\frac{\pi}{3}\right)+\left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ Now, $$\frac{11 \pi}{12} = \frac{\pi}{3} + \left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ represents the sum of two known angles.

Step 2: Apply the cosine sum formula

Using the cosine sum formula, we have: $$\cos\left(\frac{11 \pi}{12}\right) = \cos\left(\frac{\pi}{3} + \left(\frac{\pi}{4} + \frac{\pi}{6}\right)\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right) - \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$

Step 3: Apply the cosine sum formula again

We still have the sum of two angles inside the cosine function, so we apply the cosine sum formula again: $$\cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{6}\right)$$ And the same for the sine function: $$\sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{6}\right)$$

Step 4: Substitute known values and simplify

We know the exact values of cosine and sine functions for standard angles, so we substitute them into our equation and simplify: $$\cos\left(\frac{11 \pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\left(\frac{\sqrt{3}}{2}\right) - \frac{\sqrt{2}}{2}\left(\frac{1}{2}\right)\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\left(\frac{\sqrt{3}}{2}\right) + \frac{\sqrt{2}}{2}\left(\frac{1}{2}\right)\right)$$ Simplify the expression: $$\cos\left(\frac{11 \pi}{12}\right) = -\frac{\sqrt{2}+\sqrt{6}}{4}$$ So, the exact value of $$\cos\left(\frac{11 \pi}{12}\right)$$ is $$-\frac{\sqrt{2}+\sqrt{6}}{4}$$.

Key Concepts

Cosine Sum Formula

The Cosine Sum Formula is a powerful trigonometric identity that allows us to find the cosine of the sum of two angles. It states that for any angles \( A \) and \( B \), the formula is given by:

• \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

This formula is handy when dealing with non-standard angles which can be broken down into a combination of standard angles. In this exercise, we used the formula to find \( \cos \left( \frac{\pi}{3} + \frac{\pi}{4} + \frac{\pi}{6} \right) \) by simplifying it into a manageable expression. Understanding and being able to manipulate these formulas is key to mastering more complex trigonometric problems.

Standard Angles

Standard angles are specific angles for which the trigonometric values are known precisely without needing a calculator. In radians, these commonly used angles include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), etc.
For these angles, the cosine and sine values are widely used in trigonometry due to their easily recognizable exact values.

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

In our solution, the angle \( \frac{11 \pi}{12} \) was broken down into the sum of standard angles so their known values could be employed with the sum formulas.

Exact Trigonometric Values

Exact trigonometric values are crucial for solving problems without needing numerical approximations. Especially for the angles \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \), these values can be memorized:

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

Having these values at your fingertips allows you to substitute them directly into trigonometric identities and simplify calculations. As they are based on common angles on the unit circle, identifying these relationships is foundational for understanding more complex trigonometry operations.

Angle Addition Formula

The Angle Addition Formula is essential for breaking down more complex angles into sums of simpler, more manageable angles. In context, these formulas help find trigonometric values of non-standard angles:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

By breaking \( \frac{11 \pi}{12} \) into \( \frac{\pi}{3} + \frac{\pi}{4} + \frac{\pi}{6} \), we are able to apply these formulas and make complex calculations deal with step-by-step solutions. Mastering the Angle Addition Formula will significantly enhance your ability to handle trigonometric problems across various disciplines.