Problem 19 - Algebra and Trigonometry

Question

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$ 2 \cos ^{2} \frac{\pi}{8}-1 $$

Step-by-Step Solution

Answer

The exact value of the expression \(2 \cos ^{2} \frac{\pi}{8}-1\) is \(\frac{\sqrt{2}}{2}\).

Step 1: Recognize the double-angle formula in given expression

We recognize that \(2 \cos ^{2} \frac{\pi}{8}-1\) can be rewritten using the formula \(2 \cos^2{x} - 1 = \cos{2x}\), where \(x = \frac{\pi}{8}\). This formula is a rearranged form of the double angle formula for cosine.

Step 2: Convert into double-angle form

Applying the double cosine formula, the given expression is equivalent to \( \cos 2 \left(\frac{\pi}{8}\right) = \cos\left(\frac{\pi}{4}\right)\), as \(2 \cdot \frac{\pi}{8} = \frac{\pi}{4}\).

Step 3: Evaluate the exact value

We know from standard values of trigonometric functions that \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). This is the exact value of the given expression.

Key Concepts

Double-Angle Formulas

In trigonometry, double-angle formulas are essential for expressing trigonometric functions at double their original angle. These formulas help simplify complex expressions and solve equations more efficiently. Specifically for the cosine function, the double-angle formula is:

• \( \cos(2x) = 2 \cos^2(x) - 1 \)

This formula is derived from the Pythagorean identity and helps relate the trigonometric values at \(2x\) to those at \(x\). By recognizing patterns in expressions, such as \(2 \cos^2(\frac{\pi}{8}) - 1\), we can apply the double-angle formula to simplify the expression and find exact values more easily. Double-angle formulas are widely used in integration, solving trigonometric equations, and simplifying expressions in physics and engineering.

Cosine Function

The cosine function is one of the primary trigonometric functions and is often symbolized as \(\cos(x)\). It describes the x-coordinate of a point on the unit circle as the angle \(x\) varies, forming a cosine wave. This function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians. Cosine values range from -1 to 1. Key points include:

• \(\cos(0) = 1\)
• \(\cos(\pi/2) = 0\)
• \(\cos(\pi) = -1\)
• \(\cos(3\pi/2) = 0\)
• \(\cos(2\pi) = 1\)

The cosine function is even, meaning \(\cos(-x) = \cos(x)\). This symmetry makes it particularly useful in various mathematical applications. Understanding the behavior of the cosine function is crucial for using double-angle formulas effectively. For example, knowing that \(\cos(\pi/4) = \sqrt{2}/2\) helps in determining exact values, as in the exercise.

Exact Trigonometric Values

Exact trigonometric values are pivotal when solving trigonometric equations or simplifying expressions. These are known values that can be derived from the unit circle or memorized for key angles. For instance, angles like \(\pi/6\), \(\pi/4\), and \(\pi/3\) have well-defined trigonometric values:

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

In this exercise, to find the exact value of \(\cos(\pi/4)\), knowing its value as \(\sqrt{2}/2\) simplifies solving the problem. These exact values are derived from trigonometric identities and relationships within the unit circle, which produce consistent and reliable results in calculations. They are especially important in calculus, physics, and engineering for precise computations.