Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-10: Find the exact values. (a) \(\cos \frac{\pi}{4}+\cos \frac{\pi}{6}\) (b) \(\cos \frac{5 \pi}{12}\left(\right.\) use \(\left.\frac{5 \pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}\right)\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{2} + \sqrt{3}}{2}\); (b) \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

Step 1: Recall the Cosine Values

First, we need to remember the exact values of cosine for standard angles. We have:\[\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \quad \text{and} \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\]

Step 2: Add the Cosine Values

To solve part (a), simply add the cosine values we just found:\[\cos \frac{\pi}{4} + \cos \frac{\pi}{6} = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\]

Step 3: Simplify the Expression

Since both terms are fractions with the same denominator, we can combine them:\[\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{2} + \sqrt{3}}{2}\]

Step 4: Use Cosine Addition Formula

For part (b), use the cosine angle addition identity: \[\cos(a + b) = \cos a \cos b - \sin a \sin b\]We treat this as \(a = \frac{\pi}{4}\) and \(b = \frac{\pi}{6}\).

Step 5: Substitute Known Values

Substitute the known cosine and sine values:\[\cos \frac{5\pi}{12} = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \]Calculate each term:\[\cos \frac{5\pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\]

Step 6: Combine Terms

Combine the terms to find the result:\[\cos \frac{5\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}\]

Key Concepts

cosine angle addition identity

The cosine angle addition identity is a fundamental trigonometric formula used when you need to find the cosine of the sum of two angles. This identity is expressed as:

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

This helps break down complex angles into more manageable standard angles, which are known values. By using basic trigonometric identities and known angle values, you can solve problems more efficiently. This formula is incredibly useful when encountering angles that aren't typical, allowing you to express them in terms of known angles. By applying this identity, you simplify calculations for sums of angles like in part (b) of the exercise, where \( \frac{5\pi}{12} \) is expressed as \( \frac{\pi}{4} + \frac{\pi}{6} \). Gathering the sine and cosine values for these individual angles simplifies finding the result.

exact trigonometric values

Exact trigonometric values are set values for trigonometric functions (sine, cosine, tangent) at specific angles. These standard angles typically include \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), among others. These are important because they provide reliable benchmarks when calculating trigonometric functions in problems involving geometry and calculus.
For instance, in this exercise, it's crucial to know that:

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

These values are frequently used in trigonometry, allowing for precise calculations without a calculator. Memorizing these exact values simplifies the process of solving trigonometric expressions, as seen in part (a) of the exercise, where these exact values are directly added.

standard angles

Standard angles refer to commonly used angles in trigonometry. They're like a set of reference points that help simplify the calculations of trigonometric problems. Typically, these angles are 0, 30, 45, 60, and 90 degrees, which correspond to \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) in radians.
Understanding and using standard angles enables you to apply specific known trigonometric values accurately. This becomes extremely useful in exercises like these where you are given tasks to find compound angles using their standard components, such as expressing \(\frac{5\pi}{12}\) as a sum of \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\). This representation plays into the method of breaking down unfamiliar angles into combinations of familiar ones, thus simplifying the solving process.

cosine addition formula

The cosine addition formula, similar to the cosine angle addition identity, is used to find the cosine of the sum of two angles. It is expressed as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). By utilizing this formula, you can transform complex angle calculations into simpler parts by breaking them into standard angles with known trigonometric values.
For instance, when calculating \( \cos \frac{5\pi}{12} \), you express it in terms of \( \frac{\pi}{4} \) and \( \frac{\pi}{6} \). After that, substitute the known cosine and sine values:

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

This allows you to express \( \cos \left(\frac{\pi}{4} + \frac{\pi}{6}\right) \) as the combination of these values, yielding the solution \( \frac{\sqrt{6} - \sqrt{2}}{4} \). The formula is a powerful tool in trigonometry, enabling the solution of angles not directly measurable from standard angles.