Problem 86 - Algebra and Trigonometry

Question

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of x for which both sides are defined but not equal. \(\sin x-\sin x \cos ^{2} x=\sin ^{3} x\)

Step-by-Step Solution

Answer

Graph both sides, \(\sin x \cdot \sin^2x\) and \(\sin^3x\), of the equation. If the two graph lines do not coincide, then the equation is not an identity. Also, check for equality for different x-values. If for any x-value, the two sides are not equal, then it's confirmed that the equation isn't an identity. Hence, to disprove that it's an identity, it's enough to find just one x-value for which both sides of the equation are not equal.

Step 1: Analyze Both Sides of The Equation Separately

On the left, we have \(\sin x-\sin x \cos ^{2} x\) and on the right \(\sin ^{3} x\). We need to simplify each side as much as possible. On the left side, we can factor out \(\sin x\), and it becomes \(\sin x(1-\cos^2x)\). Since the identity \(\sin^2x+\cos^2x=1\), we can rewrite the expression as \(\sin x \cdot \sin^2x\).\nOn the right side, there's no need for any further simplification as it is given as \(\sin^3x\).

Step 2: Graphing Both Sides of The Equation

Graph both \(\sin x \cdot \sin^2x\) and \(\sin^3x\) using a graphing utility. Choose an appropriate viewing rectangle to observe the nature of the graphs. If the graphs look identical, it points towards the fact that the equation might be an identity. However, if the graphs do not coincide, this indicates the equation is not an identity.

Step 3: Check for Eqality for Different x-values

To establish whether the equation is an identity, examine appropriate values of x. If for every valid x-value both sides of the equation are equal, it confirms that the equation is indeed identity. However, if you find even just one valid x-value for which both sides of the equation do not equal, it shows that the equation is not an identity.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions helps in visualizing the equation to check if both sides match across different values of \(x\). Here, we have two expressions: \(\sin x \cdot \sin^2 x\) and \(\sin^3 x\). By graphing these equations together, we can compare their behavior easily.

• Use graphing software or a calculator to plot both expressions.
• Select an appropriate viewing rectangle so any potential intersection or non-matching areas are visible.
• If the graphs align perfectly for all \(x\) values, the equation may be an identity.
• Discrepancies in the graphs mean the equation is not an identity.

Graphing provides a visual approach to confirm mathematical identities and detect equivalence or differences.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions can reveal hidden relationships or simplify complex equations. Consider the left side of the equation, \(\sin x - \sin x \cos^2 x\). By factoring out \(\sin x\), you get \(\sin x(1 - \cos^2 x)\).

• Use trigonometric identities: \(\sin^2x + \cos^2x = 1\) simplifies \(1 - \cos^2 x\) to \(\sin^2 x\).
• The expression on the left side simplifies to \(\sin x \cdot \sin^2 x\), or simply \(\sin^3 x\).
• This matches the right side of the equation, compact as \(\sin^3 x\).

Simplifying both sides of the equation might show they are identical, indicating a stronger case for the equation being an identity.

Verifying Identities

When verifying trigonometric identities, the goal is to confirm that both sides of the equation produce the same result. Here’s how to approach it:

• First, simplify both sides of the equation separately, reducing them to their simplest forms.
• Check if both simplified forms match or can be transformed into one another using trigonometric identities.
• If they are identical algebraically, it confirms the equation as an identity.

Graphical verification by plotting both sides is another strategy. If they superimpose perfectly over a continuous domain, it further supports the identity claim. This multi-step approach strengthens your understanding and validation process.

Trigonometric Equations

Trigonometric equations involve solving for values of \(x\) that make the equation true. Examining conditions under which an equation holds is crucial:

• Investigate each side of the equation for different values of \(x\), especially common angles like 0, \(\frac{\pi}{2}\), \(\pi\), etc.
• If there exists at least one \(x\) that does not satisfy the equation, it is not an identity.
• Trigonometric equations can be periodic, meaning they repeat values in a regular cycle. Examine this aspect for all potential solutions.

By evaluating specific \(x\) values and considering possible non-solutions, you can determine if an equation consistently holds true or if exceptions exist.