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Chapter 5: Problem 57

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$2 \tan ^{2} x+7 \tan x-15=0$$

Short Answer

Expert verified
The approximate solutions are \(x_1 = 0.931\), \(x_2 = 3.073\), \(x_3 = 4.487\), and \(x_4 = 6.630\), all within the interval \([0,2\pi)\), and to three decimal places.

Step by step solution

01

Rewrite the equation in standard quadratic form

Use the quadratic formula \(x = (-b\pm \sqrt{b^2 - 4ac}) / 2a\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. Here, our equation can be written as \(2y^2 + 7y - 15 = 0\), where \(y = \tan x\). So, \(a = 2\), \(b = 7\), and \(c = -15\).
02

Find the roots of the equation

Substitute the values of \(a\), \(b\), and \(c\) in the quadratic formula to find the roots \(y_1\) and \(y_2\). We obtain \(y_1 = 1.5\) and \(y_2 = -5\). So, we have two equations \(\tan x = y_1\) and \(\tan x = y_2\), or \(\tan x = 1.5\) and \(\tan x = -5\).
03

Solve for x using the graph

Plot the function \(y = \tan x\) and the values \(y = 1.5\) and \(y = -5\) on a graphing calculator. Check the intervals where the graph intersects with \(y = 1.5\) and \(y = -5\). Those x-values (in the interval \([0,2\pi)\)) will be the solutions of our equation.
04

Approximate to three decimal places

Using the calculator's built-in tools or approximations, find the approximate x-values of these intersections to three decimal places. Each intersection represents a solution to the original equation.

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Most popular questions from this chapter

a sharpshooter intends to hit a target at a distance of 1000 yards with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the gun's minimum angle of elevation \(\theta\) if the range \(r\) is given by $$r=\frac{1}{32} v_{0}^{2} \sin 2 \theta$$

The table shows the average daily high temperatures in Houston \(H\) (in degrees Fahrenheit) for month \(t,\) with \(t=1\) corresponding to January. (Source: National Climatic Data Center) $$ \begin{array}{|c|c|} \hline \text { Month, } t & \text { Houston, } \boldsymbol{H} \\ \hline 1 & 62.3 \\ 2 & 66.5 \\ 3 & 73.3 \\ 4 & 79.1 \\ 5 & 85.5 \\ 6 & 90.7 \\ 7 & 93.6 \\ 8 & 93.5 \\ 9 & 89.3 \\ 10 & 82.0 \\ 11 & 72.0 \\ 12 & 64.6 \\ \hline \end{array} $$ (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above \(86^{\circ} \mathrm{F}\) and below \(86^{\circ} \mathrm{F}\).

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$12 \sin ^{2} x-13 \sin x+3=0$$

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Solve the multiple-angle equation. $$\sin 2 x=-\frac{\sqrt{3}}{2}$$
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