/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Solve each equation in part (a) ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 20

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\sin x=\sin 2 x\) (b) \(\sin x>\sin 2 x\)

Short Answer

Expert verified
From \(\sin x = \sin 2x\), solutions are \(x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}\). For \(\sin x > \sin 2x\), analyze graph and find intervals.

Step by step solution

01

Understand the Equation

We need to solve the equation \( \sin x = \sin 2x \) over the interval \([0, 2\pi)\). This equation involves trigonometric functions, and we are looking for all solutions where both sides of the equation are equal.
02

Apply the Double Angle Formula

Recall that \( \sin 2x = 2 \sin x \cos x\). Substituting this into the equation gives us \( \sin x = 2 \sin x \cos x \).
03

Factor and Solve

Rearrange the equation to \( \sin x - 2 \sin x \cos x = 0 \). Factor out \( \sin x \), resulting in \( \sin x (1 - 2 \cos x) = 0 \). Now, set each factor to zero: \( \sin x = 0 \) or \( 1 - 2 \cos x = 0 \).
04

Solve \( \sin x = 0 \)

The general solution for \( \sin x = 0 \) in \([0, 2\pi)\) is \( x = n\pi \), where \( n \) is an integer. Within the given interval, we have \( x = 0, \pi\).
05

Solve \( 1 - 2 \cos x = 0 \)

Simplifying gives \( \cos x = \frac{1}{2} \). The solutions for \( \cos x = \frac{1}{2} \) on \([0, 2\pi)\) are \( x = \frac{\pi}{3}, \frac{5\pi}{3} \).
06

Combine Solutions

The solutions to the equation \( \sin x = \sin 2x \) on \([0, 2\pi)\) are \( x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3} \).
07

Analyze the Inequality

To solve \( \sin x > \sin 2x \), graph both \( \sin x \) and \( \sin 2x \) over \([0, 2\pi)\), and find the intervals where \( \sin x \) lies above \( \sin 2x \).
08

Interpret the Graph

From the graph, analyze the intervals between the points found in the equation \( \sin x = \sin 2x \). Check the behavior of the trigonometric functions between \( x = 0, \frac{\pi}{3}, \pi, \frac{5\pi}{3} \) to determine where \( \sin x > \sin 2x \).

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Key Concepts

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

Analytical Methods
Analytical methods are essential in solving mathematical problems by relying on algebraic manipulations and previously established formulas and identities. For trigonometric equations, this means cleverly applying known identities and logical steps to break down complex expressions. The goal is to find all the exact solutions over a specified interval without relying on numerical approximation.
When solving the equation \( \sin x \ = \sin 2x \, \) we aim to find angles \( x \) that satisfy this equation within the interval \([0, 2\pi)\). This involves manipulating the equation using trigonometric identities and further techniques, such as factoring, to isolate potential solutions.
Double Angle Formula
The double angle formula is a potent tool in trigonometry, specifically useful when dealing with expressions like \( \sin 2x \). The formula given by \( \sin 2x = 2 \sin x \cos x\) allows us to express a function with a doubled angle in terms of single angles.
This transformation helps simplify and solve equations by converting seemingly complex trigonometric terms into more manageable expressions.
When we substitute this formula into the initial equation \( \sin x = \sin 2x \), we transform it into \( \sin x = 2 \sin x \cos x \). This reveals opportunities to apply further algebraic techniques, such as factoring.
Factoring Equations
Factoring is a powerful algebraic technique used to simplify and solve equations. By rewriting an equation as a product of its factors, one can easily identify solutions by considering when each factor equals zero.
After applying the double angle formula and rearranging terms in the equation \( \sin x = 2 \sin x \cos x \), we have \( \sin x (1 - 2 \cos x) = 0 \). This equation can be factored into two simple equations:
  • \( \sin x = 0 \)
  • \( 1 - 2 \cos x = 0 \)
By setting each factor to zero, we find solutions that meet the main equation's criteria within the specified interval.
Graphical Inequalities
Graphical methods provide a visual approach to solving inequalities. By plotting functions on a graph, we can easily see where one function exceeds another.
In the inequality \( \sin x > \sin 2x \), we graph both \( \sin x \) and \( \sin 2x \) over the interval \([0, 2\pi)\,\) and then examine where \( \sin x \) is above \( \sin 2x \).
This visual method clearly shows the intervals of interest. Observing how the sine wave of \( \sin x \) intersects or stays above that of \( \sin 2x \) helps identify exact ranges where the inequality holds true.

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Most popular questions from this chapter

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\operatorname{arccot} \frac{\sqrt{4-u^{2}}}{u}\right)$$

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Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(2 \sqrt{3} \sin 2 x=\sqrt{3}\) (b) \(2 \sqrt{3} \sin 2 x \leq \sqrt{3}\)

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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?
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