Problem 112 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\sin ^{3} \theta+\cos ^{3} \theta=(\cos \theta+\sin \theta)(1-\cos \theta \sin \theta)\)

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Understand the Given Equation

We are given the equation \(\sin^3 \theta + \cos^3 \theta = (\cos \theta + \sin \theta)(1 - \cos \theta \sin \theta)\). Our task is to show this equation is an identity, meaning it holds true for all values of \(\theta\).

Step 2: Apply the Sum of Cubes Formula

Recognize that \(\sin^3 \theta + \cos^3 \theta\) is similar to the sum of cubes formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Substitute \(a = \sin \theta\) and \(b = \cos \theta\) to rewrite as: \(\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\).

Step 3: Simplify Using a Trigonometric Identity

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the expression: \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta)\).

Step 4: Compare Both Sides

Notice both sides are now in the form \((\sin \theta + \cos \theta)(1 - \cos \theta \sin \theta)\). This means the original and transformed equations are identical.

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is a useful algebraic tool that provides a way to simplify expressions involving the cubes of two terms. The formula is expressed as \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \). This can be of great help when trying to verify identities or solve equations where cubic terms are present.

In the context of the given trigonometric identity, the formula helps to rewrite \( \sin^3 \theta + \cos^3 \theta \) in a more manageable format. By setting \( a = \sin \theta \) and \( b = \cos \theta \), we can transform \( \sin^3 \theta + \cos^3 \theta \) into:

• First, substitute: \( a = \sin \theta \), \( b = \cos \theta \)
• Apply the formula: \((\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\)

This transformation simplifies the verification process of the identity by providing an easy comparison to the right side of the equation.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental trigonometric identities. It states that in any angle \( \theta \), the square of the sine of that angle plus the square of the cosine of the same angle is equal to one: \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity is pivotal because it connects the two most basic trigonometric functions—sine and cosine—in a simple equation. It also allows us to simplify expressions and solve equations more efficiently.

In verifying our trigonometric identity, the Pythagorean identity helps reduce the expression \( \sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta \). By acknowledging that \( \sin^2 \theta + \cos^2 \theta = 1 \), we can rewrite the expression within parentheses as \( 1 - \sin \theta \cos \theta \). This crucial substitution aligns with the equation's right side, proving consistency and verifying the identity.

Trigonometric Equations

Trigonometric equations involve expressions with trigonometric functions that must be solved for specific angles or verified as identities. These equations can often be complex due to the cyclical nature of trigonometric functions and their periodic properties.

In solving or verifying trigonometric equations, various techniques come into play, including:

• Simplifying expressions with known identities, like the Pythagorean identity, to reduce complexity.
• Recognizing patterns or forms, such as the sum of cubes, to transform and compare expressions.
• Adjusting the expressions to equate both sides of the equation to recognize their equivalence.

The primary goal, as seen in our original task of verifying \( \sin^3 \theta + \cos^3 \theta = (\cos \theta + \sin \theta)(1 - \cos \theta \sin \theta) \), is to ensure the equation holds true across all permissible values of \( \theta \). Successfully proving this demonstrates not only the correctness of the given identity but also an understanding of how trigonometric equations operate and can be managed.