Problem 70 - Algebra and Trigonometry

Question

Use \(\cos \frac{5 \pi}{12}=\cos \left(\frac{\pi}{6}+\frac{\pi}{4}\right)\) and the formula for the cosine of the sum of two angles to find the exact value of \(\cos \frac{5 \pi}{12}\) (Section \(6.2,\) Example 4 )

Step-by-Step Solution

Answer

\(\cos \frac{5 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}\)

Step 1: Use the sum of angles formula for cosine

Apply the formula \(\cos (A+B) = \cos A \cos B - \sin A \sin B\) to \(\cos \left(\frac{\pi}{6}+\frac{\pi}{4}\right)\). This will give: \(\cos \left(\frac{\pi}{6}+\frac{\pi}{4}\right) = \cos \frac{\pi}{6} \cos \frac{\pi}{4} - \sin \frac{\pi}{6} \sin \frac{\pi}{4}\).

Step 2: Substitute the sine and cosine values

Substitute the following known values: \(\cos \frac{\pi}{6} = \sqrt{3}/2\), \(\cos \frac{\pi}{4} = \sqrt{2}/2\), \(\sin \frac{\pi}{6} = 1/2\), and \(\sin \frac{\pi}{4} = \sqrt{2}/2\). The equation will then become: \(\cos \frac{5 \pi}{12} = \left(\sqrt{3}/2\right) \left(\sqrt{2}/2\right) - \left(1/2\right) \left(\sqrt{2}/2\right)\).

Step 3: Simplify the equation

After simplifying, the equation will be as follows: \(\cos \frac{5 \pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\). Combining the similar terms gives : \(\cos \frac{5 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}\)

Key Concepts

Sum of Angles Formula

Understanding the sum of angles formula is a cornerstone in trigonometry, enabling us to find the cosine of an angle that is made up of two other angles. Simply put, this formula allows us to break down complex problems into more manageable pieces.

The sum of angles formula for cosine is expressed as \( \cos(A+B) = \cos A \cos B - \sin A \sin B \). This identity is used when you have an angle \(A+B\) and you need to express its cosine in terms of the cosines and sines of \(A\) and \(B\) individually. It serves as a powerful tool for calculating exact values of trigonometric functions for angles not found in common trigonometric tables.

How does it help in our exercise? By recognizing that \( \frac{5\pi}{12} \) can be expressed as a sum \( \frac{\pi}{6} + \frac{\pi}{4} \), we can apply the sum of angles formula to find the exact value of \(\cos \frac{5\pi}{12}\) without needing a calculator. This method also builds a deeper understanding of the relationships between angles and their trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities simplify expressions and solve trigonometric equations. Some of the most basic identities include reciprocal, Pythagorean, and even-odd identities.

The sum of angles formula itself is a trigonometric identity. In our exercise, we use this identity to work out the value of \(\cos \frac{5\pi}{12}\) without having to memorize a vast number of trigonometric values. Equipped with these identities, one can derive various trigonometric values beyond the standard 30°, 45°, and 60°, significantly expanding our capability to solve trigonometry problems.

Studying these identities does more than just help solve problems; it enables students to understand the inherent properties of trigonometric functions and their interrelations, which is vital for more advanced studies in mathematics and sciences.

Exact Trigonometric Values

Exact trigonometric values refer to the precise values of trigonometric functions — sine, cosine, and tangent — for specific angles, often expressed in terms of square roots and fractions rather than decimals. Knowing these values is crucial since they lay the foundation for solving trigonometry problems without a calculator and for understanding how trigonometric functions behave over different intervals.

In our exercise, we substitute the known exact trigonometric values for \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\), where \(\frac{\pi}{6}\) corresponds to 30° and \(\frac{\pi}{4}\) corresponds to 45°. These are part of a special set of angles called 'special angles' for which exact values are often memorized.

By using exact values such as \(\frac{\sqrt{3}}{2}\) for \(\frac{\pi}{6}\) and \(\frac{\sqrt{2}}{2}\) for \(\frac{\pi}{4}\), one delves into the beauty of mathematics where precision meets simplicity. Mastering these values enhances understanding and speeds up problem-solving, saving time on tests and building confidence for any student.