Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \frac{7 \pi}{8} \cos \frac{\pi}{8}+\sin \frac{7 \pi}{8} \sin \frac{\pi}{8}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(-\frac{\sqrt{2}}{2}\).

Step 1: Identify the Expression

The given expression is \( \cos \frac{7\pi}{8} \cos \frac{\pi}{8} + \sin \frac{7\pi}{8} \sin \frac{\pi}{8} \). We need to use trigonometric identities to find its exact value.

Step 2: Recall the Cosine Angle Sum Identity

The identity \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) is useful here. Notice that our expression matches the right side of this identity.

Step 3: Write the Expression as a Cosine of a Sum

Using the identity from Step 2, rewrite the expression as \( \cos \left( \frac{7\pi}{8} - \frac{\pi}{8} \right) \).

Step 4: Simplify the Argument of the Cosine Function

Calculate \( \frac{7\pi}{8} - \frac{\pi}{8} \). This simplifies to \( \frac{7\pi}{8} - \frac{\pi}{8} = \frac{6\pi}{8} = \frac{3\pi}{4} \).

Step 5: Evaluate the Cosine Function

Now we will find \( \cos \frac{3\pi}{4} \). Since \( \frac{3\pi}{4} \) is in the second quadrant, and cosine is negative in the second quadrant, \( \cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} \). Since \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), we have \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).

Key Concepts

Cosine Angle Sum Identity

To solve trigonometric problems involving the sum or difference of angles, we use the Cosine Angle Sum Identity. This identity states:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Understanding these identities allows us to simplify complex trigonometric expressions. In the original exercise, the expression \( \cos \frac{7\pi}{8} \cos \frac{\pi}{8} + \sin \frac{7\pi}{8} \sin \frac{\pi}{8} \) directly matches the pattern of \( \cos(A - B) \). This shows the power of these identities—they can transform a long expression into a simpler form. Here, recognizing the expression fits the identity brings us to write it as a single cosine term: \( \cos \left( \frac{7\pi}{8} - \frac{\pi}{8} \right) \). This move condenses our task into calculating the cosine of a simpler angle.

Exact Value Calculation

Calculating the exact value of trigonometric expressions without a calculator requires deep understanding of angle values and trigonometric functions. Once the expression was reduced using the Cosine Angle Sum Identity, the problem becomes finding the exact value of \( \cos \frac{3\pi}{4} \). To do this, one must be familiar with the values of trigonometric functions at standard angles:

• The cosine of \( \frac{\pi}{4} \) is a well-known value, \( \frac{\sqrt{2}}{2} \).
• \( \cos \frac{3\pi}{4} \) is found by considering its reference angle \( \frac{\pi}{4} \).

Since \( \frac{3\pi}{4} \) is in the second quadrant, where cosine values are negative, the exact value of \( \cos \frac{3\pi}{4} \) is \(-\frac{\sqrt{2}}{2} \). Knowing these reference values is crucial for evaluating trigonometric expressions accurately.

Quadrant Analysis

Understanding which quadrant an angle is in is crucial for determining the sign of its trigonometric functions. Angles measured in radians like \( \frac{3\pi}{4} \) require us to analyze their position on the unit circle.Let's break it down:

• Divide the unit circle into four quadrants.
• Quadrant I: Angles from \(0\) to \(\frac{\pi}{2}\) where all trigonometric functions are positive.
• Quadrant II: Angles from \(\frac{\pi}{2}\) to \(\pi\) where sine is positive, and cosine and tangent are negative.
• Quadrant III and IV follow similar rules based on their range.

Since \( \frac{3\pi}{4} \) is greater than \( \frac{\pi}{2} \) but less than \( \pi \), it lies in the second quadrant. Thus, cosine is negative here, and this aligns with our calculation that \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \). Knowing the properties of each quadrant helps correctly determine the signs of trigonometric values.