Problem 126 - Precalculus with Limits

Question

In Exercises 125 - 128, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. \( \sin 4\beta = 4 \sin \beta \cos \beta\left(1 - 2 \sin^2 \beta\right) \)

Step-by-Step Solution

Answer

The given equation is indeed a trigonometric identity as verified both algebraically and graphically. The algebraic proof uses the formula for the quadruple angle of sine and the double angle formulas of sine and cosine

Step 1: Properties of Sine and Cosine

Remember that \( \sin^2 x + \cos^2 x = 1 \) is a fundamental trigonometric identity. Therefore, we can rearrange the term \( \cos^2 x \) as \( 1 - \sin^2 x \) if required.

Step 2: Apply Quadruple Angle Formula

First, write down the formula for the quadruple angle of sine, \( \sin 4x = 2 \sin 2x \cos 2x \). Inside the formula, write down the double angle formulas for \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = 1 - 2 \sin^2 x \). Substituting these into the quadruple angle formula, we get \( \sin 4x = 4 \sin x \cos x (1 - 2 \sin^2 x) \).

Step 3: Graphical Verification

To check visually if the two functions are equal, plot the left side (\( y = \sin 4\beta \)) and the right side (\( y = 4 \sin \beta \cos \beta(1 - 2 \sin^2 \beta) \)) of the given trigonometric identity using a graphing utility. If the functions are identical, their graphs would coincide, hence confirming the identity graphically.

Key Concepts

Double Angle Formulas

When studying trigonometry, one basic principle to comprehend is the Double Angle Formulas. These are vital because they tell us how to simplify expressions involving twice the angle.

For sine, the double angle formula is expressed as:
\[ \sin(2x) = 2 \sin(x) \cos(x) \].
This formula arises from the sum formula for sine and it can be used in various trigonometric simplifications and proofs. We can also find the double angle formulas for cosine and tangent, but the one for sine is especially relevant when expanding to other, larger angle formulas such as the quadruple angle formula.

Quadruple Angle Formula

Expanding our toolbox further, we encounter the Quadruple Angle Formula, which allows for expressions involving four times a given angle to be simplified. The quadruple angle formula for sine can be constructed by applying the double angle formula twice.

This looks like:
\[ \sin(4x) = 2\sin(2x)\cos(2x) \].
This is further expanded by substituting the double angle formulas for \(\sin(2x)\) and \(\cos(2x)\) to give us an expression in terms of \(\sin(x)\) and \(\cos(x)\), as seen in the solution for our exercise. Such formulas are incredibly useful in complex trigonometric problems and proofs.

Graphing Utility Verification

A modern and powerful approach to verify trigonometric identities is the use of a graphing utility. This tool provides a visual representation of trigonometric functions and can be a quick way to confirm identities.

When entered correctly, if the two sides of a supposed identity produce the same graph, it confirms that the identity holds for all angles in the domain of the functions. This method can serve as a tangible confirmation of an identity, which is particularly helpful when algebraic manipulations become overwhelming.

Sine and Cosine Properties

Understanding the basic properties of sine and cosine functions is essential for any student of trigonometry. Both functions are periodic, with sine and cosine having a period of \(2\pi\) radians or 360 degrees. This means they repeat their values every \(2\pi\) radians.

Additionally, they're bound between -1 and 1 and are foundational to describing circular motion and oscillations. The most fundamental identity connecting them is \(\sin^2(x) + \cos^2(x) = 1\), often referred to as the Pythagorean identity. Understanding this relationship is crucial when simplifying trigonometric expressions and proving various trigonometric identities, as demonstrated in the step-by-step solution for our exercise.