Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{2} x-\frac{1}{2}$$

Step-by-Step Solution

Answer

The given expression \( \cos ^{2}x-\frac{1}{2} \) can be rewritten using double-angle formula as \( \frac{\cos 2x}{2} \). Both expressions are identical as verified by graphing.

Step 1: Identify double-angle identity for cosine

The double-angle identity for cosine is given by \( \cos 2x = 2\cos^2{x}-1\). We can rearrange this identity to make it match the given expression \( \cos^2{x} - \frac{1}{2} \).

Step 2: Rearrange the double-angle identity

Rearranging the double-angle identity \( \cos 2x = 2\cos^2{x}-1 \), we get \( \cos^2{x} = \frac{\cos 2x+1}{2} \). Subtracting 0.5 from both sides, we get \( \cos ^{2}x-\frac{1}{2}=\frac{\cos 2x}{2} \). Then this is the rewritten form of the given expression using double-angle formula.

Step 3: Graphing to verify the equality

By graphing the original expression \( \cos^2{x} - \frac{1}{2} \) and the rewritten expression \( \frac{\cos 2x}{2} \) using a graphing utility, we can visually verify that both expressions indeed describe the same function.

Key Concepts

Cosine Double-Angle Identity

The Cosine Double-Angle Identity is a fundamental trigonometric formula used to simplify expressions involving angles. It states that for any angle \( x \), the identity is given by:\[\cos 2x = 2\cos^2{x} - 1\]This identity allows us to express the cosine of a double angle in terms of the cosine of a single angle. By rearranging this formula, we can solve for different forms which are useful in rewriting complex trigonometric expressions.
In the problem, we manipulated this identity to rewrite \( \cos^2 x - \frac{1}{2} \). We first solve for \( \cos^2 x \) to find \( \cos^2 x = \frac{\cos{2x} + 1}{2} \). Subtracting \( \frac{1}{2} \) from this gives us the expression \( \frac{\cos 2x}{2} \). This step shows how versatile double-angle identities are in transforming trigonometric problems.

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions like sine, cosine, and tangent. They hold true for all angles and help in simplifying expressions and solving trigonometric equations. Familiarity with these identities extends your problem-solving toolkit for a wide range of mathematical problems.There are several types of identities, but the double-angle formulas are particularly handy when dealing with expressions of angles that are multiples of a base angle. For example:

• Sum and Difference Formulas
• Double Angle Formulas
• Half Angle Formulas

Applying these identities, such as the cosine double-angle identity, simplifies calculations and makes it easier to verify the equivalence of different mathematical expressions.
In this exercise, the goal was to leverage the cosine double-angle identity to rewrite and verify that \( \cos^2 x - \frac{1}{2} \) and \( \frac{\cos 2x}{2} \) are equivalent. With practice, these identities become invaluable tools in trigonometry.

Graphing Trigonometric Functions

Graphing trigonometric functions visually demonstrates how these functions behave over different angles. By using a graphing utility, you can plot equations to observe their periodic nature and any symmetries they possess.In this exercise, graphing played a crucial role in validating that both expressions \( \cos^2{x} - \frac{1}{2} \) and \( \frac{\cos{2x}}{2} \) are indeed the same. By observing their graphs, which overlap perfectly, we confirmed that these algebraic manipulations held true graphically as well.
When learning about trigonometric identities, using graphs can enhance understanding and provide confirmation of equivalences in a way that's easy to grasp.

• Graphs of sine, cosine, and tangent functions illustrate their characteristic waves.
• Comparing graphs can help visualize identities and verify transformations.
• Graphing utilities offer an interactive way to explore mathematical functions.

This visual tool is particularly helpful in education, allowing students to bridge the gap between abstract algebraic expressions and their geometric representations, making it easier to comprehend the consistent behaviors of trigonometric functions.