Problem 10 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\(- \frac{\sqrt{2+\sqrt{3}}}{2}\).

Step 1: Break Down the Angle

Notice that the angle \( \frac{13\pi}{12} \) can be rewritten as the sum of two angles whose cosine values we know. Let’s express \( \frac{13\pi}{12} \) as the sum of \( \pi \) and \( \frac{\pi}{12} \). Thus, \( \frac{13\pi}{12} = \pi + \frac{\pi}{12} \).

Step 2: Apply the Cosine Addition Formula

Using the angle addition identity, \( \cos(a + b) = \cos a \cos b - \sin a \sin b \), substitute \( a = \pi \) and \( b = \frac{\pi}{12} \). Therefore, \( \cos(\pi + \frac{\pi}{12}) = \cos \pi \cos \frac{\pi}{12} - \sin \pi \sin \frac{\pi}{12} \).

Step 3: Use Known Trigonometric Values

We know that \( \cos \pi = -1 \), \( \sin \pi = 0 \). Thus, \( \cos(\pi + \frac{\pi}{12}) = (-1) \cos \frac{\pi}{12} - 0 \cdot \sin \frac{\pi}{12} = -\cos \frac{\pi}{12} \). Now, focus on finding \( \cos \frac{\pi}{12} \).

Step 4: Use Half-Angle Identity

To find \( \cos \frac{\pi}{12} \), let's use the half-angle identity: \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}} \). We choose \( \theta = \frac{\pi}{6} \) so that \( \frac{\pi}{12} = \frac{\pi}{6} \times \frac{1}{2} \). Then \( \cos \frac{\pi}{12} = \sqrt{\frac{1 + \cos \frac{\pi}{6}}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \).

Step 5: Simplify the Expression

Simplify the expression for \( \cos \frac{\pi}{12} \): \( \cos \frac{\pi}{12} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2+\sqrt{3}}}{2} \). Substitute this value back to find \( \cos \left( \frac{13\pi}{12} \right) = -\cos \frac{\pi}{12} = - \frac{\sqrt{2+\sqrt{3}}}{2} \).

Key Concepts

Cosine Addition Formula

The cosine addition formula is a powerful tool in trigonometry. It helps us find the cosine of the sum of two angles. The formula is expressed mathematically as:

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

This identity is useful for manipulating trigonometric expressions and finding precise values without a calculator.
In the given problem, this formula was used to break down the angle \(\frac{13\pi}{12}\) into more familiar angles, \(\pi\) and \(\frac{\pi}{12}\).
The values \(\cos \pi = -1\) and \(\sin \pi = 0\) were then substituted into the formula to simplify further. By handling the addition of angles this way, we could move closer to finding an exact expression for \(\cos\left(\frac{13\pi}{12}\right)\). Understanding this formula opens up the way to approximating and understanding more complicated angle measures in trigonometry.

Half-Angle Identity

The half-angle identity is a part of a set of identities in trigonometry that helps in finding the exact values for angles that aren't directly found on typical trigonometric charts. The half-angle identity for cosine is given by:

• \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}}\)

This formula can produce two results, represented by the \(\pm\) sign. The sign is chosen based on the quadrant in which the angle \(\frac{\theta}{2}\) lies.
In this problem, finding \(\cos \frac{\pi}{12}\) involved applying the identity with \(\theta = \frac{\pi}{6}\), yielding the expression \(\sqrt{\frac{2 + \sqrt{3}}{4}}\).
This half-angle identity thus aids in breaking down the trigonometric function into more manageable parts, thereby assisting in finding exact trigonometric values for angles like \(\frac{\pi}{12}\) which aren't typically found on standard unit circle measurements.

Trigonometric Values

Trigonometric values for specific angles are a fundamental aspect of trigonometry. Knowing the exact values for angles such as \(0\), \(30\), \(45\), \(60\), and \(90\) degrees (or their radian equivalents) is crucial.
In this problem, recognizing that \(\cos \pi = -1\) and \(\sin \pi = 0\) allows simplification before applying additional identities.
The trigonometric values for certain angles are frequently used to either directly solve problems or to work through formulas like the cosine addition formula or the half-angle identity. By being familiar with these basic values, students can tackle harder trigonometry problems using identities, as demonstrated in the solution. Learning these values by heart enables quicker calculation and deeper understanding of angle measures and their trigonometric functions.