Problem 96 - 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. $$4 \cos ^{2} \frac{x}{2}=2+2 \cos x$$

Step-by-Step Solution

Answer

The graphs and the nature of the equation will determine whether this equation is an identity or not. If the equation is an identity, both graphs will coincide. Otherwise, the two graphs will differ for at least one value of \(x\), which is the counter-example.

Step 1: Break Down The Equation

We can transform the given equation \(4 \cos ^{2} \frac{x}{2}=2+2 \cos x\) to be suitable for graphical analysis. By implementing double-angle formula, we can get \(\cos ^{2} \frac{x}{2} = \frac{1 + \cos x}{2}\) and multiplying each side by 4 yields the same form as our original equation, \(4 \cos ^{2} \frac{x}{2}=2+2 \cos x\).

Step 2: Graph The Various Parts of The Equation

Now graph both \(y=4 \cos ^{2} \frac{x}{2}\) and \(y=2+2 \cos x\) using a graphing calculator. Make sure each side of the equation is graphed in the same viewing rectangle.

Step 3: Analyze The Graphs

If the graphs completely overlap, the equation is an identity. If they do not coincide, find a value of \(x\) for which both sides are defined but not equal.

Step 4: Verify or Disprove Identity

Proving the equation is an identity needs a direct verification which results in a true statement for all values of \(x\). Otherwise, to disprove it, find a single counter-example i.e., a value of \(x\) for which the equation becomes false.

Key Concepts

Double-Angle Formula

The double-angle formula is a cornerstone in trigonometry that expresses trigonometric functions of doubled angles in terms of functions of the original angle. It specifically allows for simplification and transformation of trigonometric expressions. For cosine, the double-angle formula is: \[ \cos 2\theta = 2\cos^2 \theta - 1 \]. In the exercise, we encounter the term \( \cos^2 \frac{x}{2} \), which can be transformed using a form of the double-angle formula: \[ \cos^2 \frac{x}{2} = \frac{1 + \cos x}{2} \]. This identity allows us to rewrite and simplify the given trigonometric equation so that we can perform graphical analysis more effectively. By recognizing and applying the appropriate formula, you halve the angle and express it in a way that's compatible with other expressions, all of which is crucial for graph-based verifications.

Graphical Analysis

Graphical analysis of equations is a visual method used in mathematics to understand and solve equations by looking at their graphs. For trigonometric equations, this often involves graphing each part of the equation separately and determining where they intersect or coincide. To apply graphical analysis to our problem, you will create plots for \( y = 4 \cos^2 \left( \frac{x}{2} \right) \) and \( y = 2 + 2 \cos x \).

• Use a graphing calculator or software to plot both functions.
  
• Ensure both are displayed within the same viewing rectangle to compare their graphs accurately.
  
• If both graphs completely overlap for all values within a reasonable domain, this suggests the equation is an identity.

Overlapping graphs indicate that both expressions are equal over the entire range of x, making the equation a plausible identity. If not, you might need to identify particular x-values where discrepancies occur.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and solutions may require algebraic manipulation and familiarity with trigonometric identities and properties. Solving such equations often involves finding particular angles or verifying identities. In our exercise, we are tasked with determining whether the equation \( 4 \cos^2 \frac{x}{2} = 2 + 2 \cos x \) is a trigonometric identity. This process involves using known

• trigonometric identities to transform one side of the equation to match the other,
  
• graphical analysis to verify or refute the equivalence over an interval.

The equation can be manipulated using identities, then graphed to visually assess its validity across the real number spectrum. If an identity is not apparent, a single counterexample might be sought to show that the equation does not hold universally, thus it is not an identity.