Problem 7 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12} $$

Step-by-Step Solution

Answer

The values \(\alpha = \frac{5\pi}{12}\) and \(\beta = \frac{\pi}{12}\) can be found from the given expression. It can then be written in the form of \(\cos(\frac{5\pi}{12} - \frac{\pi}{12})\), which simplifies to \(\cos(\frac{\pi}{3})\). Thus, the exact value of the expression is \(\frac{1}{2}\).

Step 1: Identify \(\alpha\) and \(\beta\)

In the given expression \(\cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}\), the angles \(\alpha\) and \(\beta\) can be identified by looking at the terms that are multiplied with the sine and cosine functions. Thus, \(\alpha = \frac{5\pi}{12}\) and \(\beta = \frac{\pi}{12}\) .

Step 2: Write the expression as the cosine of an angle

Taking \(\cos(\alpha - \beta)\) = \(\cos\alpha \cos\beta + \sin\alpha \sin\beta\) and substituting \(\alpha = \frac{5\pi}{12}\) and \(\beta = \frac{\pi}{12}\) into the formula, the expression can be written as \(\cos(\frac{5\pi}{12} - \frac{\pi}{12})\)

Step 3: Calculate the exact value of the expression

Now the expression \(\cos(\frac{5\pi}{12} - \frac{\pi}{12})\) can be simplified to \(\cos(\frac{4\pi}{12})\), this further simplifies to \(\cos(\frac{\pi}{3})\). Using trigonometric values, this can be found to be exactly \(\frac{1}{2}\)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle measures. They are very useful for simplifying expressions and solving equations in trigonometry.
One common identity is the tangent of the difference of two angles, but in this exercise, we focus on the cosine of the difference. Understanding these identities allows you to express complex trigonometric terms in simpler forms.

• The basic functions, such as sine, cosine, and tangent, are defined using the unit circle.
• Identities like the Pythagorean identity, sum and difference formulas, and double-angle formulas are fundamental tools.
• The cosine difference identity is an important formula in trigonometry, allowing us to write expressions like \(\cos(\alpha - \beta)\) in terms of \(\cos\alpha\), \(\cos\beta\), \(\sin\alpha\), and \(\sin\beta\).

These identities help in solving equations and simplifying expressions. Mastering them provides a deeper understanding of the relationships between angles and sides in right triangles.

Exact Trigonometric Values

Exact trigonometric values are specific values of trigonometric functions that can be precisely calculated for certain angles. These angles are typically found at notable positions on the unit circle, such as \(\frac{\pi}{3}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{6}\).
Knowing these exact values is crucial because they provide a reference for solving problems without the need for a calculator.

• The cosine of \(\frac{\pi}{3}\) is one of these exact values and equals \frac{1}{2}\.
• The sine of \(\frac{\pi}{3}\) is \frac{\sqrt{3}}{2}\, showcasing another important trigonometric value.
• Using the unit circle, these values can be derived geometrically, offering clear visualizations of their origins.

These values simplify complex calculations, reducing the risk of errors due to rounding or approximation, thus they are vital in trigonometric exercises.

Angle Subtraction Formula

The angle subtraction formula is a specific trigonometric identity used to find the cosine or sine of the difference between two angles. Here, we are focusing on the cosine version, which is especially useful in simplifying expressions like the one in this exercise.
To apply this formula accurately, consider the following key elements:

• The formula is \(\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta\).
• This identity helps convert complicated products like the expression given in the exercise to a simple cosine of a single angle.
• Using the formula, you can reduce the problem of finding trigonometric values from combined angles to evaluating a single angle, making it much more straightforward.

This formula not only simplifies expressions but also aids in solving various types of trigonometric problems, making it an essential tool in any trigonometry toolkit.