Problem 112 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sin 4 \beta=4 \sin \beta \cos \beta\left(1-2 \sin ^{2} \beta\right)$$

Step-by-Step Solution

Answer

The original identity \[\sin 4 \beta = 4 \sin \beta \cos \beta(1-2 \sin^{2} \beta)\] is verified through a series of transformations using trigonometric identities. Additionally, the identity is confirmed through graphical representation.

Step 1: Understanding the identities

Recall the double angle formula for sine: \[ \sin 2x = 2\sin x \cos x \]. Also understand that \[ \sin^2x = 1 - \cos^2x \] (Pythagorean identity), which is the key to link sine and cosine with a quadratic relationship.

Step 2: Apply the double angle formula

When looking at the right hand side of the equation, we start breaking it down with the double angle formula of sine. Replace \[4 \sin \beta \cos \beta\] with \[\sin 2\beta\]. Now we have: \[\sin 2\beta(1-2 \sin ^{2} \beta)\]

Step 3: Apply the Pythagorean identity

We can further use the Pythagorean identity on \[\sin ^{2} \beta\] inside the parenthesis. It can be rewritten as \[1 - \cos^{2} \beta\], making the whole equation look like this: \[\sin 2\beta[1-2(1 - \cos^{2} \beta)]\] which simplifies to \[\sin 2\beta[2\cos^{2}\beta - 1]\]

Step 4: Apply the double angle formula again

Now apply the double angle formula for cosine: \[ \cos 2x = 2\cos^{2}x - 1 \]. The expression in brackets can be rewritten as \[\cos 2\beta\], so the final expression becomes \[\sin 2\beta \cos 2\beta\].

Step 5: Apply the double angle formula one last time

Employing the double angle formula for sine one last time yields the expression on the left. So, \[ \sin (2 \times 2\beta) = \sin 4\beta \] and the identity is verified.

Step 6: Graphing

Using a graphing utility, plot both expressions to visually verify that they are the same. They should produce the same graph, further verifying the identity.

Key Concepts

Double Angle Formula

The Double Angle Formula is a valuable tool in trigonometry, allowing us to express trigonometric functions of doubled angles in terms of single angles. For sine, the double angle formula is given by

• \[\sin 2x = 2 \sin x \cos x \]

This formula reveals that the sine of an angle twice as large as the original can be dissected into the product of sine and cosine of the single angle, multiplied by 2.
In verifying trigonometric identities, the double angle formulas are often employed to simplify the expressions or transform them into different, yet equivalent forms. In the provided exercise, applying this formula helped in reducing complexity by replacing \(4 \sin \beta \cos \beta\) with \(\sin 2\beta\). This strategic use is essential because it simplifies the manipulation of the expression and facilitates the further application of other trigonometric identities.

Pythagorean Identity

The Pythagorean Identity is another cornerstone of trigonometry, connecting the sine and cosine functions through a squared relationship. It is expressed as

• \[\sin^2 x + \cos^2 x = 1\]

This identity is incredibly useful when trying to express one trigonometric function in terms of another. By rearranging this identity, one can express \(\sin^2 x\) as

• \(1 - \cos^2 x\)

In the exercise, the identity was leveraged to rewrite \(\sin^2 \beta\) and further simplify the given expression. This conversion of sine in terms of cosine helps in algebraic manipulations, especially when the goal is to use another form in subsequent verification steps.

Graphing Utility

While algebraic verification is rigorous, using a graphing utility provides a visual method of verifying trigonometric identities. A graphing utility can plot functions based on point values and visually display their behavior over a specified domain.
For our problem, plotting both sides of the identity in a graphing utility involves:

• Inputting the left-hand function \(\sin 4\beta\)
• Using the expression from the right-hand side \(4 \sin \beta \cos \beta(1-2 \sin^2 \beta)\)

By observing that the graphs overlap perfectly over the domain, you visually confirm they are equivalent, thus strengthening the algebraic proof. This approach not only confirms correctness but also aids in understanding the behavior of the trigonometric relationships graphically.

Algebraic Simplification

Algebraic Simplification is the process of rewriting complex mathematical expressions in a simpler, more manageable form. This is often done by identifying and applying various algebraic rules or identities.
In the context of trigonometric identities, simplification involves techniques like factoring, combining like terms, or recognizing trigonometric identities to reduce expressions.

• For example, the identity \( \cos 2x = 2\cos^2 x - 1 \) was used in the exercise to replace parts of the expression, making it easier to handle.
• The expression \(\sin 2\beta[2\cos^2\beta - 1]\) was successfully transformed into \(\sin 2\beta \cos 2\beta\), which we could further simplify using known identities.

Stripping down the expression to its simplest form is crucial for both clarity and application of other trigonometric properties, which is especially beneficial in verifying complex identities.