Problem 53 - Precalculus: A Concise Course

Question

Use a graph to solve the equation on the interval \([-2 \pi, 2 \pi]\). $$ \sec x=-2 $$

Step-by-Step Solution

Answer

The solutions to the equation \(\sec x = -2\) on the interval \([-2π, 2π]\) are \(x = \frac{2π}{3}\), \(x = \frac{4π}{3}\), \(x = \frac{8π}{3}\), and \(x = \frac{10π}{3}\).

Step 1: Translate secant to cosine

The secant of x is the reciprocal of the cosine of x. Hence, the given equation \(\sec x = -2\) is translated or converted to \(\cos x = -\frac{1}{2}\).

Step 2: Refer the cosine graph

Refer to the graph of the function \(\cos x\). Look for the points where \(\cos x = -\frac{1}{2}\).

Step 3: Identify the interval points

The x-values of these points of intersection on the \(\cos x\) graph are the solutions to the equation \(\sec x = -2\) on the interval \([-2π, 2π]\). Identify these points.

Step 4: Final solution

The solutions are \(x = \frac{2π}{3}\), \(x = \frac{4π}{3}\), \(x = \frac{8π}{3}\), and \(x = \frac{10π}{3}\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is an important concept in trigonometry and is closely related to the cosine function. The secant of an angle is defined as the reciprocal of the cosine of that angle. This means that:

• \( \sec x = \frac{1}{\cos x} \)

When you see an equation involving \( \sec x \), it's useful to translate it into \( \cos x \) because most people are more familiar with cosine.

In our exercise, \( \sec x = -2 \) translates to \( \cos x = -\frac{1}{2} \). This translation helps us simplify the problem using properties and graphs of the cosine function.

Understanding the secant function as the reciprocal of cosine is critical because it allows us to leverage the well-known characteristics of the cosine graph to find solutions to equations involving secant.

Cosine Function

The cosine function, denoted as \( \cos x \), is a fundamental trigonometric function that shows the relationship of an angle in a right-angled triangle. For the unit circle, \( \cos x \) represents the x-coordinate of a point as it moves around the circle.

Key properties of the cosine function:

• The cosine function has a period of \( 2\pi \), meaning it repeats every \( 2\pi \).
• Cosine values range between -1 and 1.
• \( \cos x \) is an even function, so \( \cos(-x) = \cos x \).

In this problem, we look for \( x \) such that \( \cos x = -\frac{1}{2} \). The equation suggests points on the unit circle where the x-coordinate (cosine value) equals \(-\frac{1}{2}\). Knowing cosine behaviors makes it easier to find solutions within specific intervals, like \( [-2\pi, 2\pi] \).

There are established angles where cosine takes specific values, which helps in identifying solutions.

Graphical Solution

Solving trigonometric equations using graphs is both a visual and analytical approach. For our exercise \( \sec x = -2 \), first translate it to \( \cos x = -\frac{1}{2} \). Then, focus on the graph of the cosine function.

• Plot \( \cos x \) over the desired domain, which is \([-2\pi, 2\pi]\) in this case.
• Identify where the graph intersects the horizontal line \( y = -\frac{1}{2} \).

These intersection points provide the x-values that solve the equation.

Specifically, when examining a cosine graph, you can find how the wave dips to \(-\frac{1}{2}\) at specific points. Within the interval \([-2\pi, 2\pi]\), these solutions include intersections at \( x = \frac{2\pi}{3} \), \( x = \frac{4\pi}{3} \), \( x = \frac{8\pi}{3} \), and \( x = \frac{10\pi}{3} \).

Graphical solutions are often intuitive as they allow us to see where specific conditions are met and how periodicity plays a role. This method offers a deeper understanding of the trigonometric functions and their relationships.