Problem 38 - Precalculus with Limits

Question

In Exercises 37 - 42, find the exact value of the expression. \( \cos \dfrac{\pi}{16} \cos \dfrac{3\pi}{16} - \sin \dfrac{\pi}{16} \sin \dfrac{3\pi}{16} \)

Step-by-Step Solution

Answer

The exact value of the given expression is \( \sqrt{2+\sqrt{2}}/2 \).

Step 1: Identify and Apply the Appropriate Identitiy

Upon observing the expression, it resembles the formula for cosine of sum of two angles. The formula is defined as \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). If the given expression is assessed from this perspective, A=B=\(\frac{\pi}{16}\) and C=2A. Hence, this can be written as \( \cos \dfrac{\pi}{8}\) (which is sum of the two angles to be considered.

Step 2: Simplify

Since the range of cosine goes from -1 to 1, and \(\dfrac{\pi}{8}\) is in the first quadrant (where cosine is positive), the answer is confirmed as a value between 0 and 1.

Step 3: Use unit circle for final calculation

This step involves extracting the final solution based on principles from the unit circle. The final result obtained will be \( \cos \dfrac{\pi}{8} = \sqrt{2+\sqrt{2}}/2 \). This involves steps of how cosine of 45, 30, 60 degrees and their multiples are derived using the unit circle and Pythagoras theorem.

Key Concepts

Cosine of Sum of Angles

To understand the problem, let's explore the concept of cosine of sum of angles. The trigonometric identity for the cosine of the sum of two angles is a nifty tool: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] This identity helps simplify expressions involving cosine, as seen in the original exercise. Here, the expression \( \cos \frac{\pi}{16} \cos \frac{3\pi}{16} - \sin \frac{\pi}{16} \sin \frac{3\pi}{16} \) directly fits the formula. By identifying \( A \) and \( B \) in the formula as both \( \frac{\pi}{16} \), this expression simplifies to \( \cos(\frac{\pi}{8}) \). Using this identity allows us to find the exact trigonometric value more easily.

• Recognize patterns that match known identities.
• Substitute the appropriate angle values.
• Simplify using known trigonometric expansions.

Unit Circle

The unit circle is a powerful visual and analytical tool for understanding trigonometric functions. It's a circle with a radius of one centered at the origin of the coordinate plane, with angles measured in radians. The unit circle helps us determine the values of trigonometric functions at various angles. The coordinates of any point on the unit circle are represented as \((\cos \theta, \sin \theta)\). This means if you have an angle \( \theta \), the x-coordinate gives you \( \cos \theta \), and the y-coordinate gives you \( \sin \theta \). In our exercise, once we simplified the expression to \( \cos \frac{\pi}{8} \), we can use the unit circle to find this value precisely:

• Locate the angle \( \frac{\pi}{8} \) on the unit circle.
• Determine the cosine value by checking the x-coordinate.

This steamlines finding precise, rational results without drawing the entire circle for each problem.

Angle Measurement in Radians

Angles can be measured in degrees or radians, but in advanced mathematics, radians are a preferred measure. This is because radians relate directly to the unit circle. In terms of radians, a full circle is \( 2\pi \). To convert degrees to radians, we use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \] Our exercise uses angles like \( \frac{\pi}{16} \) and \( \frac{3\pi}{16} \), highlighting the precision papers gain when working with radians.

• Radians allow easy reference to the unit circle.
• Make transitions between trigonometric identities seamless.
• Enable calculations to remain consistent and straightforward.

For the simplification process, grasping this radiantic representation aids in smoother, error-free computation.