Chapter 10.3, Problem 42PS

Solution

To find: The interval in which cosine function increases or secant decreases.

The cosine function increases in $\left(\pi ,2\pi \right)$ and decreases in interval $\left(2\pi ,2.5\pi \right)$ and the sec function is absent in the interval $\left(\pi ,1.5\pi \right)$ , decreases in interval $\left(1.5\pi ,2\pi \right)$ and increases in interval $\left(2\pi ,2.5\pi \right)$

Given data:

The cosine function increases or secant decreases.

Method/Formula used:

Graph the functions sec (x) and cos ( x ) and observe the change in both functions.

Calculation:

The graphs functions of sec( x ) and cos ( x ) are shown in Fig.1.

  

From the Fig, 1, it is observed:

In the interval $x=\pi$ to $x=1.5\pi$ the cosine function increases and sec function is absent.

In the interval $x=1.5\pi$ to $x=2\pi$ cosine function increases and sec function decreases.

In the interval $x=2\pi$ to $x=2.5\pi$ cosine function decreases and sec function increases.

This periodicity is maintained in the interval $\left(-\infty ,\infty \right)$ .