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Chapter 10: Problem 80

How many solutions does the equation \(2 x-1=0\) have? How many solutions does the equation \(2 \sin x-1=0\) have? Explain.

Short Answer

Expert verified
The equation \(2x - 1 = 0\) has one solution; \(2\sin x - 1 = 0\) has infinitely many solutions.

Step by step solution

01

Identify the Equation Type

The first equation is a simple linear equation, while the second equation involves a trigonometric function. This tells us that their solutions are determined differently.
02

Solve the Linear Equation

The equation is \(2x - 1 = 0\). To find \(x\), add 1 to both sides and then divide by 2: \(2x = 1\), so \(x = \frac{1}{2}\). Since a linear equation has exactly one solution, \(2x - 1 = 0\) has one solution.
03

Understand the Trigonometric Equation

The second equation is \(2\sin x - 1 = 0\). It can be rewritten as \(\sin x = \frac{1}{2}\). The sine function is periodic with a period of \(2\pi\), and each cycle has two solutions for \(\sin x = \frac{1}{2}\).
04

Find General Solutions for the Trigonometric Equation

We can determine the solutions where \(\sin x = \frac{1}{2}\) within one period \([0, 2\pi)\). The standard angles are \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\). Since the sine function is periodic, every complete cycle (each \(2\pi\)) these solutions repeat.
05

Determine Number of Solutions

The linear equation \(2x - 1 = 0\) has only one solution: \(x = \frac{1}{2}\). The trigonometric equation \(2\sin x - 1 = 0\) has infinitely many solutions because the sine function repeats these two solutions \(x = \frac{\pi}{6} + 2k\pi\) and \(x = \frac{5\pi}{6} + 2k\pi\) for any integer \(k\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are equations of the first degree, and they represent straight lines when graphed. A linear equation typically takes the form \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the unknown variable you need to solve for. Solving them is a straightforward process, as they usually have exactly one solution, provided \(a\) is not zero. This solution can be found by isolating \(x\) on one side of the equation, typically through basic algebraic operations like addition, subtraction, multiplication, or division.
  • For example, given the equation \(2x - 1 = 0\), we isolate \(x\) by first adding 1 to both sides, resulting in \(2x = 1\).
  • We then divide both sides by 2, giving us the solution \(x = \frac{1}{2}\).

This simplicity and straightforward nature are the hallmark of linear equations, compromising their unique solution in most cases.
Trigonometric Functions
Trigonometric functions relate to angles and can describe waves and oscillations. One of the most common trigonometric functions is the sine function, denoted as \(\sin x\). The sine function is particularly interesting because it oscillates between -1 and 1, creating a wave-like shape. When solving trigonometric equations like \(2\sin x - 1 = 0\), we first isolate the trigonometric function.
  • This equation simplifies to \(\sin x = \frac{1}{2}\), a classic angle value in trigonometry.
  • Within a standard period \([0, 2\pi)\), there are specific angles where \(\sin x = \frac{1}{2}\), which are typically \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\).

Understanding trigonometric functions is essential for fields like physics and engineering, where wave patterns frequently occur.
Periodicity
Periodicity is a defining characteristic of trigonometric functions. It refers to the repeating pattern or cycle in the graph of the function. For the sine function, the periodicity is \(2\pi\), meaning that every \(2\pi\) units or radians, the graph repeats its shape and value sequence.
  • In the equation \(2\sin x - 1 = 0\), this periodicity means that the solutions \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\) will repeat every \(2\pi\) radians.
  • This gives us an infinite number of solutions, expressed generally as \(x = \frac{\pi}{6} + 2k\pi\) and \(x = \frac{5\pi}{6} + 2k\pi\), where \(k\) is any integer.

Periodicity is crucial in understanding not only simple trigonometric functions but also more complex wave interactions and cycles seen in nature and technology.

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Most popular questions from this chapter

Use a calculator to find each value. $$\cot (\arccos 0.58236841)$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos x=\sin ^{2} \frac{x}{2}$$

Determining Wattage Amperage is a measure of the amount of electricity that is moving through a circuit, while voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.) (a) A household circuit has voltage $$V=163 \sin 120 \pi t$$ when an incandescent light bulb is turned on with amperage $$I=1.23 \sin 120 \pi t$$ Graph the wattage $$W=V I$$ that is consumed by the light bulb over the interval \(0 \leq t \leq 0.05\) (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to find values for \(a, c,\) and \(\omega\) so that $$ W=a \cos \omega t+c $$ (d) Check your answer in part (c) by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph from part (a) to estimate the average watt. age used by the light. How many watts do you think this incandescent light bulb is rated for?

Hearing Different Tones When a musical instrument creates a tone of \(110 \mathrm{Hz}\), it also creates tones at \(220,330\) \(440,550,660, \dots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration, but it can reproduce the higher frequencies, called the upper harmonics. The low tones can still be heard, because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. We can model this phenomenon with a graphing calculator. (a) In the window \([0,0.03]\) by \([-1,1]\), graph the upper harmonics represented by the pressure $$ \begin{aligned} P=& \frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t] \\ &+\frac{1}{4} \sin [2 \pi(440) t] \end{aligned} $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above in the window \([0,0.03]\) by \([-2,2]\) (Image can't copy)

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\csc ^{4} x-\cot ^{4} x=\frac{1+\cos ^{2} x}{1-\cos ^{2} x}$$
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