Problem 125 - Precalculus with Limits

Question

In Exercises 125 - 128, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. \( \cos 3\beta = \cos^3 \beta - 3 \sin^2 \beta \cos \beta \)

Step-by-Step Solution

Answer

The provided equation \( \cos 3\beta = \cos^3 \beta - 3 \sin^2 \beta \cos \beta \) is a correct trigonometric identity as it transforms into the standard form of \(\cos 3\beta\) identity. This has been verified using a graphing utility and algebraic manipulations.

Step 1: Graph the given equation

Using a graphing utility, plot both sides of the equation, \( \cos 3\beta \) and \( \cos^3 \beta - 3 \sin^2 \beta \cos \beta \), as two separate functions. If they are equal, the two graphs would overlap completely.

Step 2: Transform the left-hand side equation

The left-hand side \(\cos 3\beta\) is already in simplest form, so no transformation is needed.

Step 3: Transform the right-hand side equation using the Pythagorean identity

We can substitute \( \sin^2 \beta \) in the right-hand side equation with \(1 - \cos^2 \beta \) by using the Pythagorean identity \(\sin^2x + \cos^2x = 1\). So, the equation could be written as : \( \cos^3 \beta - 3 (1 - \cos^2 \beta) \cos \beta \).

Step 4: Simplify the right-hand side equation

Upon simplifying, the equation is \( \cos^3 \beta - 3\cos \beta + 3\cos^3 \beta \), which results into \(4\cos^3 \beta - 3\cos \beta\).

Step 5: Recognize the obtained equation

The final equation \(4\cos^3 \beta - 3\cos \beta\) is a well-known form of \(\cos 3\beta\) trigonometric identity, thus indicating that the initial equation holds true.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It's extensively used to solve problems involving triangles and to model periodic phenomena such as waves. One fundamental skill in trigonometry is understanding and manipulating trigonometric functions like sine, cosine, and tangent.

For example, the cosine function, denoted as \( \text{cos} \), relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In the context of the exercise at hand, \( \text{cos} \) plays a crucial role in verifying trigonometric identities and transforming equations.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry which states that for any angle \( x \), the sum of the squares of sine and cosine of \( x \) is always equal to one, expressed as \( \text{sin}^2x + \text{cos}^2x = 1 \).

This identity comes from the Pythagorean theorem, which is about the sides of a right triangle. The Pythagorean identity is incredibly useful for transforming trigonometric expressions, which often must be simplified or manipulated to solve equations and prove other identities, as we can see in the textbook exercise.

Graphing Utility

A graphing utility is an important tool for visualizing mathematical equations, particularly in trigonometry. These utilities, which can be software programs or graphing calculators, allow users to plot graphs of functions and equations quickly.

The utility plays a vital role in verifying trigonometric identities, as seen in Step 1 of the textbook solution. By graphing both sides of the given identity, we can visually confirm if they are equivalent. If one side of the identity matches the other when plotted, it supports the claim that the two expressions are, in fact, the same.

Algebraic Verification

Algebraic verification is the process of proving that a given trigonometric identity holds true by transforming and simplifying the equation through algebraic manipulation. This process often involves utilizing fundamental identities, like the Pythagorean identity, to rewrite trigonometric functions in terms of others.

In the presented exercise, the algebraic verification is shown through Steps 2 to 5. Simplifying the given expression algebraically helps confirm the identity without depending only on graphical evidence. Thus, it's a reliable way to validate the equality of two trigonometric expressions, ensuring that the identity is true for all values within the domains of the functions involved.