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Chapter 5: Problem 49

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$2 \sin x+\cos x=0$$

Short Answer

Expert verified
The solutions of the equation \(2 \sin x+\cos x=0\) in the interval \([0,2 \pi)\), approximated to three decimal places, can be found using a graphing utility. The exact x-values for these solutions depend on the capability of the graphing utility to provide accurate and precise values.

Step by step solution

01

Setup the Equation

Begin by setting the problem up on the graphing utility. Take the equation, \(2 \sin x+\cos x=0\), and plug it into the graphing calculator's plotting interface. The y-axis represents the value of the equation for each x within the given interval \([0,2 \pi)\). Hence we need to find the x-values for which y=0.
02

Identify the Solutions

Since the task is to find the solutions of the equation, look for the points on the graph where y=0. These are the points where the curve intersects with the x-axis. Approximate the x-values of these points to three decimal places. You can do this by zooming in at the intersecting points and by using the trace function of your graphing utility.
03

Verify the Solutions

It's always a good idea to verify the obtained solutions. Input the obtained x-values back into the original equation \(2 \sin x+\cos x=0\) to ensure that they satisfy the equation.

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Most popular questions from this chapter

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sec x+\tan x-x$$ Trigonometric Equation $$\sec x \tan x+\sec ^{2} x-1=0$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=2 \sin x+\cos 2 x$$ Trigonometric Equation $$2 \cos x-4 \sin x \cos x=0$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 2 x+\tan x}{1-\tan 2 x \tan x}$$

The displacement from equilibrium of a weight oscillating on the end of a spring is given by \(y=1.56 e^{-0.22 t} \cos 4.9 t,\) where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds). Use a graphing utility to graph the displacement function for \(0 \leq t \leq 10\). Find the time beyond which the displacement does not exceed 1 foot from equilibrium.

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