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Chapter 4: Problem 54

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

Short Answer

Expert verified
The solutions to the equation on the given interval are \( x = -3\pi/4, -\pi/4, \pi/4, 3\pi/4 \).

Step by step solution

01

Understand the Secant Function

The secant function, \( \sec(x) \), is the reciprocal of the cosine function. In other words, \( \sec(x) = 1/\cos(x) \). For \( \sec(x) = 2 \), the equation becomes \( 1/\cos(x) = 2 \), which simplifies to \( \cos(x) = 0.5 \). Thus, the task is to find the value/s of \( x \) that satisfy this equation in the interval \( -2 \pi , 2 \pi \).
02

Draw the cosine graph

Plot the cosine graph on the interval \( -2\pi \) to \( 2\pi \). Note the cosine curve oscillates between -1 and 1. Identify the points on the x-axis where the cosine graph intersects y = 0.5. There will be four intersections, two in each cycle of the cosine function.
03

Identify the solutions

The points of intersection of the cosine graph and \( y = 0.5 \) give the solutions to the equation. These points are \( x = -3\pi/4, -\pi/4, \pi/4, 3\pi/4 \)

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