Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{2} 2 x$$

Step-by-Step Solution

Answer

The expression \( \cos^2 2x \) when rewritten in terms of the first power of cosine, becomes \( \frac{1 + \cos 4x}{2} \).

Step 1: Apply the power-reducing identity

The identity used here is \( \cos^2 x = \frac{1 + \cos 2x}{2} \). So, \( \cos^2 2x \) becomes \( \frac{1 + \cos 4x}{2} \).

Step 2: Simplification

There's no further simplification needed, as we already have an expression in terms of the first power of cosine.

Step 3: Graphing the expressions

Using a graphing utility, graph both expressions, \( \cos^2 2x \) and \( \frac{1 + \cos 4x}{2} \). Identical graphs would verify that the expressions are equivalent.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values within the domain of the variable. These identities play an essential role in simplifying expressions, solving trigonometric equations, and transforming one function into another. Some of the most basic and widely used identities are the Pythagorean identities, which include the one used in our example: the power-reducing identity for cosine.

The power-reducing identity allows us to turn a squared trigonometric function into an expression involving the first power, making it simpler to handle and solve. For instance, the identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \) is derived from the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) and is used to rewrite \( \cos^2 2x \) in the example as \( \frac{1 + \cos 4x}{2} \) without changing the value.

Cosine First Power

Expressions involving cosine to the first power, such as \( \cos x \) or \( \cos 4x \) as seen in our original exercise, are easier to handle both algebraically and graphically. The significance of converting higher-power trigonometric expressions into first power lies in simplifying complex calculations and making the derivatives and integration of these functions more straightforward. Functions involving cosine to the first power display a periodic and oscillatory motion which can represent many natural phenomena such as waves and vibrations.

The importance of moving to 'cosine first power' is emphasized in tasks like graphing or integration where maintaining the integrity of the function's value, while making it more comprehensible is paramount. The conversion of \( \cos^2 2x \) to \( \frac{1 + \cos 4x}{2} \) is an example of using trigonometric identities to express a function in terms of the first power of cosine.

Graphing Trigonometric Functions

Graphing is a powerful tool for visualizing and understanding the behavior of trigonometric functions. It aids in confirming identities, comparing functions, and solving equations graphically. When graphing the original function \( \cos^2 2x \) and the transformed \( \frac{1 + \cos 4x}{2} \) side by side, they should overlap perfectly, demonstrating their equivalence. This graphical representation also highlights characteristics like amplitude, period, and phase shift.

By using graphing utilities, students can visually verify their work. Even if students make an algebraic mistake, a graph can often reveal discrepancies between functions. An accurate graph reflecting the periodic nature and distinctive waveform of trigonometric functions stands as an excellent learning tool and a visual proof to solidify the concept behind trigonometric identities.