Problem 27 - 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. $$6 \sin x \cos x$$

Step-by-Step Solution

Answer

The expression can be rewritten as \(3 \sin 2x\). The graphs of the initial and resulting expressions must visually overlap when plotted, confirming the identity.

Step 1: Identify the appropriate double-angle formula

In trigonometry, we have a double angle formula that relates \(sin(2x)\) to \(2 \sin x \cos x\). This can be used here.

Step 2: Apply the double-angle formula

Applying the double-angle formula, the expression \(6 \sin x \cos x\) can be rewritten as \(3 \sin 2x\).

Step 3: Verify with a graphing utility

To verify the above-formulated identity, graph both \(6 \sin x \cos x\) and \(3 \sin 2x\) using a graphing utility. You should see that the graphs of both expressions visually overlap, confirming that both expressions are equivalent.

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that focuses on studying the relationships between the sides and angles of triangles. It helps us understand how these elements interact and how to calculate unknown pieces of information in triangles. One of the key features of trigonometry is its use of trigonometric functions such as sine, cosine, and tangent. These functions allow us to model periodic phenomena and analyze relationships in a wide variety of contexts, from engineering to physics.

In the context of double-angle formulas, trigonometry becomes especially interesting because it enables us to transform and simplify complex trigonometric expressions. These transformations are not just mathematical curiosities but actually have practical applications, such as simplifying calculations or solving complex equations.

• The double-angle formula for sine is: \[ \sin(2x) = 2 \sin(x) \cos(x) \]
• Using this formula, we can rewrite expressions like \(6 \sin x \cos x\) in a simpler form \(3 \sin 2x\).
• Such rewrites are not just algebraically pleasing but also often offer deeper insights into underlying patterns and structures within mathematical problems.

As you become more familiar with these trigonometric concepts, you'll find that they are powerful tools not just in mathematics, but also in solving real-world problems.

Graphing Utility

Graphing utilities, such as graphing calculators and computer software, are invaluable tools for visualizing mathematical expressions. They allow students to see a graphical representation of the relationships described by mathematical equations.

When working with trigonometric expressions like \(6 \sin x \cos x\) and its rewritten form \(3 \sin 2x\), graphing utilities provide a visual confirmation of equivalence. By plotting both these expressions on the same set of axes, you can see the shapes and their amplitudes displayed graphically. If both expressions are indeed equivalent, their graphs will perfectly overlap.

• Graphing helps confirm algebraic manipulations visually, solidifying understanding.
• It reveals characteristics such as symmetry, periodicity, amplitude, and frequency of trigonometric functions.
• Graphs also make it easier to identify patterns and relationships that might not be immediately obvious from algebraic expressions alone.

This visual approach is a great way to verify calculations, deepen comprehension, and build confidence in handling trigonometric identities and transformations.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables, where the trigonometric functions are defined. These identities are fundamental tools in trigonometry as they allow for the simplification and manipulation of trigonometric expressions.

In this exercise, the identity used is a double-angle formula—a specialized form of trigonometric identity that relates an angle to double that angle. The specific identity \[ \sin(2x) = 2 \sin(x) \cos(x) \] enables the transformation of the expression \(6 \sin x \cos x\) into \(3 \sin 2x\), simplifying its complexity without altering its value.

• Mastering these identities allows you to transform complex expressions into simpler forms, making further calculations easier.
• The double-angle formula specifically helps reduce the number of terms in the expression, leading to streamlined solutions.
• These identities are also essential in solving trigonometric equations, proving other identities, and even in calculus for integration and differentiation purposes.

Understanding and applying trigonometric identities like double-angle formulas can significantly enhance problem-solving skills, making it easier to tackle a wide range of mathematical challenges.