Question

Use a graph to find a number \(\delta \)such that if \(\left| {x - \frac{\pi }{6}} \right| < \delta \), then \(\left| {{\rm{co}}{{\rm{s}}^2}x - \frac{3}{4}} \right| < 0.1\)

Short answer

The number is \(0.109\).

Step-by-step solution

Plot the graph of the function.

Draw the graph of the function \(f\left( x \right) = {\cos ^2}x\) by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation \({\cos ^2}X\) in the \({Y_1}\) tab.

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = {\cos ^2}x\) is shown below:

Solve the given condition

Apply the absolute property, that is, if\(\left| x \right| < a\)then\( - a < x < a\).

Rewrite the given condition \(\left| {{\rm{co}}{{\rm{s}}^2}x - \frac{3}{4}} \right| < 0.1\) as:

\(\begin{aligned}{c} - 0.1 < {\rm{co}}{{\rm{s}}^2}x - \frac{3}{4} < 0.1\\ - 0.1 + \frac{3}{4} < {\rm{co}}{{\rm{s}}^2}x < 0.1 + \frac{3}{4}\\0.65 < {\rm{co}}{{\rm{s}}^2}x < 0.85\end{aligned}\)

Solve the obtained condition for \({\bf{x}}\)

Solve the condition\({\rm{co}}{{\rm{s}}^2}x = 0.65\) and \({\rm{co}}{{\rm{s}}^2}x = 0.85\)for \(x\).

\(\begin{aligned}{c}{\rm{co}}{{\rm{s}}^2}x&= 0.65\\{\rm{cos}}x&= 0.806226\\x \approx 0.633\\\\{\rm{co}}{{\rm{s}}^2}x&= 0.85\\{\rm{cos}}x&= 0.921954\\x \approx 0.398\end{aligned}\)

Observe the graph

It is observed that the condition \(0.65 < f\left( x \right) < 0.85\) is satisfied when \(x\)satisfies \(0.398 < x < 0.633\).

From the graph, it is observed that \(\frac{\pi }{6} - {\delta _1} \approx 0.398\) and \(\frac{\pi }{6} + {\delta _2} \approx 0.633\).

Obtain the value of \({\delta _1}\)and\({\delta _2}\)

Solve the obtained condition.

\(\begin{aligned}{c}\frac{\pi }{6} - {\delta _1} \approx 0.398\\{\delta _1} \approx \frac{\pi }{6} - 0.398\\ \approx 0.126\end{aligned}\)

\(\begin{aligned}{c}\frac{\pi }{6} + {\delta _2} \approx 0.633\\{\delta _2} \approx 0.633 - \frac{\pi }{6}\\ \approx 0.109\end{aligned}\)

Obtain the value of \(\delta \)

The value of\(\delta \)is the minimum value of \({\delta _1}\), and\({\delta _2}\).

\(\begin{aligned}{c}\delta &= {\rm{min}}\left\{ {0.126,0.109} \right\}\\&= 0.109\end{aligned}\)

Therefore, the value of \(\delta \) is \(0.109\).