Problem 22 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \cos \frac{\pi}{12}-\cos \frac{5 \pi}{12} $$

Step-by-Step Solution

Answer

\sqrt{2}

Step 1: Apply the identity

The given expression is \( \cos \frac{\pi}{12}-\cos \frac{5 \pi}{12} \) which is in the form \(\cos A - \cos B\). Following the identity mentioned previously, it can be written as \( -2 \sin \left(\frac {A + B} {2}\right) \sin \left(\frac {A - B} {2}\right) \). Substituting \( A = \frac{\pi}{12} \) and \( B = \frac{5 \pi}{12} \), the expression becomes \( -2 \sin \left(\frac{\frac{\pi}{12}+\frac{5\pi}{12}}{2}\right) \sin \left(\frac{\frac{\pi}{12}-\frac{5\pi}{12}}{2}\right) \).

Step 2: Simplify the fractions

Simplifying the above expression, it transforms into \( -2 \sin \left(\frac{\pi}{4}\right) \sin \left(-\frac{\pi}{6}\right) \).

Step 3: Solve the sine functions

Knowing that \( \sin(\pi/4) = 1/\sqrt{2} \) and \( \sin(-\pi/6) = -1/2 \), substituting these values into our equation results in \( -2 * \frac{1}{\sqrt{2}} * -\frac{1}{2} \).

Step 4: Final simplification

Reducing the expression gives the final answer: \( \sqrt{2} \).

Key Concepts

Cosine Difference Identity

The cosine difference identity is useful in transforming the difference of two cosine terms into a product form which can be easier to evaluate or simplify. When you have an expression like \( \cos A - \cos B \), the identity to apply is:

• \( \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \)

This transformation is particularly helpful in simplifying expressions or solving equations where both trigonometric and algebraic reduction are involved. By rewriting the expression as a product, it often makes the evaluation more straightforward in terms of finding exact values or determining the nature of the expression.
In the problem provided, using this identity transformed the original expression \( \cos \frac{\pi}{12} - \cos \frac{5\pi}{12} \) into a form involving simpler sine calculations.

Sine Values

Understanding sine values, especially of standard angles, plays a critical role in solving problems involving trigonometric identities. Common angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) have well-known sine values:

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

For negative angles, the sine function reflects across the y-axis, meaning:\( \sin (-x) = -\sin (x) \). Knowing these sine values is essential not only for substitution purposes in identities but also for simplifying expressions to their exact trigonometric values.
For example, \( \sin (\pi/4) \) and \( \sin (-\pi/6) \) are used in the computation in our original expression, showing how critical it is to memorize and understand these basic sine values.

Angle Simplification

Angle simplification helps in making trigonometric calculations simpler. When working with trigonometric identities, it often involves manipulating angles to familiar or standard forms. In the provided solution, angles were simplified to \( \frac{\pi}{4} \) and \(-\frac{\pi}{6} \) after substitution and fraction operations. Simplifying angles often involves:

• Adding or subtracting angle measures
• Reducing fractions
• Using angle sum and difference formulas

In trigonometry, working with simpler angles -- especially those that easily translate to known sine or cosine values -- makes it possible to handle complex expressions efficiently and reach exact values in solutions.
In the exercise, simplifying both the sum \( \frac{\pi}{12} + \frac{5\pi}{12} \) to \( \frac{\pi}{4} \) and difference \( \frac{\pi}{12} - \frac{5\pi}{12} \) to \( -\frac{\pi}{6} \) illustrated this angle simplification principle.

Exact Trigonometric Values

Exact trigonometric values are those which can be represented as precise ratios of integers or known radicals rather than decimal approximations. For common angles, these values are:

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

Using exact trigonometric values not only allows for precision but also aids in verifying the correctness of a computed answer when solving problems.
In the original problem, exact values like \( \sin (\pi/4) = 1/\sqrt{2} \) and \( \sin (-\pi/6) = -1/2 \) were used to simplify an expression leading to the final exact result of \( \sqrt{2} \). Understanding exact trigonometric values is fundamental for students because it builds a strong foundation for more advanced concepts in both pure and applied mathematics.