Problem 58 - Precalculus Essentials

Question

Use a graph to solve each equation for \(-2 \pi \leq x \leq 2 \pi\) $$\sec x=1$$

Step-by-Step Solution

Answer

The solutions to the equation \( \sec x = 1 \) within the interval \(-2 \pi \leq x \leq 2 \pi\) are \( x = -2 \pi, 0, 2 \pi \).

Step 1: Rewrite secant function as reciprocal of cosine function

The trigonometric function can be rewritten as \( \cos x = 1 \) as secant is the reciprocal of the cosine function.

Step 2: Draw graphs of cos x and y=1

The next step is to graph the function \( y = \cos x \) and the line \( y = 1 \) on the same axes. The \( \cos x \) function has period \( 2 \pi \), and alternates between -1 and 1, crossing the x-axis at \( \pi / 2 + n \pi \) where \( n \) is an integer. The line \( y = 1 \) is a horizontal line that intersects the y-axis at 1.

Step 3: Find intersection points

The solutions to the equation are at the points where the graphs of \( y = \cos x \) and \( y = 1 \) intersect. Looking at the graph, these points are at \( x = -2 \pi, 0, 2 \pi \) within the given interval.

Key Concepts

Secant Function

The secant function is a fundamental trigonometric function that arises often in equations. Understanding this function involves knowing its relationship with the cosine function. Specifically, the secant function, denoted as \( \sec x \), is defined as the reciprocal of the cosine function. In mathematical terms, this can be expressed as \( \sec x = \frac{1}{\cos x} \). This reciprocal relationship implies that when the cosine of an angle is equal to one, the secant of that angle is also one. This simplifies the process of solving trigonometric equations involving the secant by transforming them into terms of the more familiar cosine function.

Cosine Function

The cosine function is one of the core trigonometric functions and is denoted as \( \cos x \). It describes the cosine of an angle, which can be visualized on the unit circle as the x-coordinate of a point. The cosine function has a distinct and important range from -1 to 1. Its graph is a smooth, continuous wave that repeats every \( 2\pi \), a characteristic known as its period (or periodicity). For any equation involving \( \sec x \), we can rewrite it in terms of \( \cos x \) because \( \sec x = \frac{1}{\cos x} \). This simplifies solutions significantly by relying on established properties of the cosine function. In this exercise, where \( \sec x = 1 \), it directly leads us to \( \cos x = 1 \) since a value of 1 for the secant implies the cosine is also 1.

Graphical Method

Using a graphical method for solving equations can provide intuitive visual recognition of solutions. For the given exercise where \( \sec x = 1 \), rewriting it as \( \cos x = 1 \) allows us to leverage the graph of the cosine function. By plotting both \( y = \cos x \) and \( y = 1 \) on the same graph, we can visually identify the points of intersection. These intersections reveal the solution to the equation.
To approach this effectively:

• Draw the graph of \( y = \cos x \).
• Include the horizontal line \( y = 1 \).
• Find intersection points within the interval \(-2\pi \leq x \leq 2\pi\).

Through this method, solutions appear at the points where \( y = \cos x \) intersects \( y = 1 \). In this exercise, you find these intersections at \( x = -2\pi, 0, 2\pi \) which can be confirmed visually by observing the graph.

Periodicity of Trigonometric Functions

Periodicity is a key property of trigonometric functions, namely the cyclical repetition of their values over specific intervals. For cosine, this period is \( 2\pi \). This means the cosine function repeats its pattern every \( 2\pi \) units along the x-axis. Periodicity is crucial when solving trigonometric equations because it gives multiple solutions within any given interval.
Understanding periodicity involves:

• Recognizing the repeating pattern of trigonometric functions. For \( \cos x \), observe that its graph peaks at \( y = 1 \), troughs at \( y = -1 \), and crosses the x-axis twice within each period.
• Identifying the repeat intervals: any solution found at \( x \) will recur every \( 2\pi \) units.
• Solving for trigonometric equations like \( \cos x = 1 \) by considering all occurrences within the specified interval.

In this exercise, periodicity helps to explain why the solutions for \( \sec x = 1 \) appear symmetrically at \( x = -2\pi, 0, 2\pi \). These are the points where the cosine wave, repeating every \( 2\pi \), reaches its maximum of 1.