Problem 53 - Precalculus: Functions and Graphs

Question

Refer to the graph of the equation on the specifled interval. Find all values of \(x\) such that for the real number \(a,\) (a) \(y=a,\) (b) \(y>a,\) and (c) \(y

Step-by-Step Solution

Answer

Values of \(x\) are: (a) \(-\frac{4\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}\); (b) \((-2\pi, -\frac{4\pi}{3}) \cup (-\frac{2\pi}{3}, \frac{2\pi}{3}) \cup (\frac{4\pi}{3}, 2\pi)\); (c) \((-\frac{4\pi}{3}, -\frac{2\pi}{3}) \cup (\frac{2\pi}{3}, \frac{4\pi}{3})\).

Step 1: Understanding the Problem

We need to find values of \(x\) for three conditions related to \(y = \cos x\), given the interval \([-2\pi, 2\pi]\) and \(a = -\frac{1}{2}\). This involves analyzing the graph of the cosine function within the specified range.

Step 2: Condition (a): Find where \(y = a\)

We set the equation \(y = \cos x = -\frac{1}{2}\). To find \(x\), we analyze the cosine function which returns \(-\frac{1}{2}\) at specific angles: \(x = \frac{2\pi}{3} + 2k\pi\) and \(x = \frac{4\pi}{3} + 2k\pi\), where \(k\) is an integer. Substituting \(k\) values gives results within the interval \([-2\pi, 2\pi]\): \(-\frac{4\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}\).

Step 3: Condition (b): Find where \(y > a\)

For \(y = \cos x > -\frac{1}{2}\), this happens between the roots found in part (a) for segments of the cosine wave above \(-\frac{1}{2}\). This gives the ranges: \((-2\pi, -\frac{4\pi}{3}) \cup (-\frac{2\pi}{3}, \frac{2\pi}{3}) \cup (\frac{4\pi}{3}, 2\pi)\).

Step 4: Condition (c): Find where \(y < a\)

For \(y = \cos x < -\frac{1}{2}\), we find segments where the cosine curve falls below \(-\frac{1}{2}\). The ranges for this are: \((-\frac{4\pi}{3}, -\frac{2\pi}{3}) \cup (\frac{2\pi}{3}, \frac{4\pi}{3})\).

Key Concepts

Properties of Trigonometric Functions

The cosine function, denoted as \(y = \cos x\), is a fundamental trigonometric function that has several important properties. Understanding these properties helps in solving equations and analyzing graphs. The cosine function is periodic, with a period of \(2\pi\), meaning its pattern repeats every \(2\pi\) radians. It is also an even function, meaning \(\cos(-x) = \cos(x)\), which implies symmetry about the y-axis.
Another crucial property of the cosine function is its range, which is from \(-1\) to \(1\). This means for any real number \(x\), \(-1 \leq \cos x \leq 1\). Its maximum value is \(1\) and occurs at integer multiples of \(2\pi\) (e.g., \(0, 2\pi, 4\pi, \ldots\)), and its minimum value is \(-1\) occurring at points one period later the maximal points like \(\pi, 3\pi, 5\pi, \ldots\).
In practice, these properties allow you to predict and solve for specific values and intervals, especially when working within defined boundaries like those set in the exercise, \([-2\pi, 2\pi]\).

Graphing Trigonometric Functions

Graphing the cosine function involves understanding its oscillating nature between \(-1\) and \(1\) over its defined interval. The graph of \(y = \cos x\) is a wave that starts at its peak value \(1\), descends to \(-1\), and returns back.

• The waveform crosses the x-axis at \(\frac{\pi}{2}, \frac{3\pi}{2},\) and similarly every \(\pi\) increment.
• It hits its maximum at \(0, 2\pi\), and minimum at \(\pi, 3\pi\) within one cycle.

To graph the cosine function effectively, it’s essential to highlight key points: maxima, minima, and intercepts, enabling the tracing of its entire period.
Marking these points accurately allows for a better understanding when adding translations or solving inequalities involving \(\cos x\). For instance, when \(a = -\frac{1}{2}\), locating where the graph equals or diverges from the line \(y = -\frac{1}{2}\) becomes straightforward.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, or tangent, set equal to a value. Solving these involves manipulating the equation to isolate the variable.

• When solving \(\cos x = a\), it is helpful to know the unit circle and values of cosine at standard angles.
• Angles where \(\cos x = -\frac{1}{2}\) are commonly \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\), plus any full rotations \(2k\pi\).

By substituting integers for \(k\) in the equation, results are found within the specified interval of the exercise. This is important because only values that lie within \([-2\pi, 2\pi]\) are initially valid solutions.
In practical scenarios, these equations help determine intersection points and intervals where the cosine graph meets specific lines or conditions.

Inequalities involving Trigonometric Functions

Inequalities like \(y > a\) or \(y < a\) with \(y = \cos x\) define regions on a trigonometric graph that lie above or below certain values. Solving these inequalities involves:

• Contrast the trigonometric function value \(y\) with the boundary values.
• For \(y = \cos x > -\frac{1}{2}\), capture sections where the graph dips less than \(-\frac{1}{2}\).

These sections are in between roots previously calculated for equations. Likewise, for \(y = \cos x < -\frac{1}{2}\), identify regions where it descends below \(-\frac{1}{2}\).
Understanding these inequalities aids in determining the precise intervals for specified conditions across multiple cycles of the cosine wave, which is essential for complete analysis of the graph within the defined boundaries.