Problem 72 - Algebra and Trigonometry

Question

\(67-72\). Find the value of the product or sum. $$ \cos \frac{\pi}{12}+\cos \frac{5 \pi}{12} $$

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{6}}{2} \).

Step 1: Utilize Cosine Addition Formula

The formula states that \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \). Let \( A = \frac{\pi}{12} \) and \( B = \frac{5\pi}{12} \).

Step 2: Calculate A+B and A-B

Now, calculate \( A + B = \frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \), and \( A - B = \frac{\pi}{12} - \frac{5\pi}{12} = -\frac{4\pi}{12} = -\frac{\pi}{3} \).

Step 3: Apply Values to Cosine Addition Formula

Substitute the calculated values into the formula: \[ \cos \frac{\pi}{12} + \cos \frac{5\pi}{12} = 2 \cos \left( \frac{\pi}{4} \right) \cos \left( -\frac{\pi}{6} \right) \] Note: \( \cos(-x) = \cos x \), thus this simplifies further to: \[ 2 \cos \left( \frac{\pi}{4} \right) \cos \left( \frac{\pi}{6} \right) \].

Step 4: Evaluate Cosine Values

Recall the value of these cosines: \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) and \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \).

Step 5: Calculate the final product

Compute the expression: \[ 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \].

Key Concepts

Cosine Addition Formula

The Cosine Addition Formula is a useful trigonometric identity that helps simplify expressions involving the sum or difference of cosines. The formula is expressed as follows: \[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]This formula allows us to rewrite the addition of two cosines in terms of the product of two other cosine functions. It becomes extremely useful when dealing with angles that can be challenging to handle directly. In the context of the exercise, we need to compute \( \cos \frac{\pi}{12} + \cos \frac{5\pi}{12} \). By using the Cosine Addition Formula, we substitute \( A = \frac{\pi}{12} \) and \( B = \frac{5\pi}{12} \). This substitution simplifies our problem by transforming it into a product, which is often easier to evaluate.

Angle Sum and Difference

The concept of Angle Sum and Difference is critical in trigonometry as it helps simplify and solve trigonometric expressions involving multiple angles. In trigonometry, calculating the sum or difference of two angles, such as \( A + B \) and \( A - B \), is a key step when using identities.

• For the sum: \( A + B = \frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \).
• For the difference: \( A - B = \frac{\pi}{12} - \frac{5\pi}{12} = -\frac{4\pi}{12} = -\frac{\pi}{3} \).

Using these calculations, the angles are substituted back into the Cosine Addition Formula, simplifying the trigonometric expression to an easier-to-evaluate form. Understanding how to deftly handle angle sum and difference is essential when manipulating trigonometric identities, as it helps to reveal intermediate values that lead to the final solution.

Trigonometric Values

Trigonometric values for standard angles are crucial for evaluating expressions involving sine, cosine, and tangent. Memorizing these standard values can greatly streamline solving trigonometric problems. For example, in the exercise, we need to know:

• The cosine of \( \frac{\pi}{4} \), which is \( \frac{\sqrt{2}}{2} \).
• The cosine of \( \frac{\pi}{6} \), which is \( \frac{\sqrt{3}}{2} \).

These values come from the standard unit circle angles and are commonly encountered in trigonometry problems. Once these known values are applied to the simplified expression \( 2 \cos \left( \frac{\pi}{4} \right) \cos \left( \frac{\pi}{6} \right) \), it becomes straightforward to compute the final result: \( \frac{\sqrt{6}}{2} \). Understanding and recalling these values is a vital step in solving problems efficiently and accurately.