Problem 95 - Precalculus Essentials

Question

In Exercises \(95-98,\) 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, find a value of \(x\) for which both sides are defined but not equal. $$3-6 \sin ^{2} x=3 \cos 2 x$$

Step-by-Step Solution

Answer

After following the outlined steps, it's found that the two sides of the equation yield the same result when graphed and evaluated at a specific \(x\) value, such as \(x=\pi\). Therefore, the equation \(3-6 \sin ^{2} x=3 \cos 2 x\) is validated as an identity.

Step 1. Transformation of the Equation into a Comparable Format

The given equation is \(3-6 \sin^2 x = 3\cos2x\). The double-angle identity for cosine function is \(\cos(2x)=1-2\sin^2(x)\). Utilizing this identity, the given equation can be transformed into: \(3-6\sin^2 x= 3(1-2\sin^2 x)\) or \(3-6\sin^2 x = 3-6\sin^2 x\). Thus, the equation, after substitution, appears to be an identity.

Step 2. Graphing the Two Sides Separately

The next step involves graphing the two sides of the equation in the same viewing rectangle. If the graphs appear to coincide completely, then the equation can be verified as an identity. For this purpose, use graphing software or a graphing calculator to plot the left side of the equation, which is \(y=3-6\sin^2 x\), and the right side of the equation, which is \(y=3\cos2x\).

Step 3. Comparing Graphs and Verifying

Once both graphs are presented in the same view, compare them carefully. If the graphs appear to coincide completely, this would suggest that the equation is indeed an identity. To further verify, check for a certain value of \(x\) if both sides of the equation corroborate. For instance, when \(x=\pi\), both sides of the equation simplify to the same value, thus confirming the equation as an identity.

Key Concepts

Double-Angle Identity

The double-angle identity is a useful tool in trigonometry for simplifying expressions and transforming functions. When dealing with the cosine function, the double-angle identity is expressed as \( \cos(2x) = 1 - 2\sin^2(x) \). This identity helps to relate the cosine of a double angle, \(2x\), with the sine squared of \(x\).

• In the original problem, the given equation \(3 - 6\sin^2 x = 3\cos 2x\) is simplified using this identity.
• By substituting \(\cos(2x)\) with \(1 - 2\sin^2(x)\), the equation becomes much more straightforward, turning into \(3 - 6\sin^2 x = 3 - 6\sin^2 x\).

This substitution confirms that the equation is an identity when both sides match after applying the double-angle identity for cosine.

Graphing Equations

Graphing equations is a powerful way to visually verify identities and understand the behavior of functions. In the problem, the expressions on both sides of the equation were graphed: the left side \(y = 3 - 6\sin^2 x\) and the right side \(y = 3\cos 2x\). When graphed in the same viewing rectangle, you can easily check if they are indeed the same.

• Graphing helps identify if the two functions coincide across their domain, which suggests that they are equivalent expressions.
• Software or a graphing calculator can be used to plot these graphs accurately, ensuring that any discrepancies can be spotted immediately.

This graphical approach ensures that the equation \(3-6\sin^2 x = 3\cos 2x\) holds true as an identity by confirming that the two graphs overlap perfectly.

Sinusoidal Functions

Sinusoidal functions involve the sine and cosine functions, which are fundamental in trigonometry, often representing periodic phenomena. In this specific problem, both \(\sin^2 x\) and \(\cos 2x\) are sinusoidal in nature, characterized by their periodic waves.

• These functions are pivotal in understanding cyclic patterns, having applications across physics, engineering, and signal processing.
• The equivalence established in the problem, \(3 - 6\sin^2 x = 3\cos 2x\), shows the interconnectedness of these functions through trigonometric identities.

Understanding how these functions relate via identities such as the double-angle identity is crucial. It not only aids in solving equations but also in predicting and modeling real-life periodic systems.