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Chapter 4: Problem 55

Use a graph to solve the equation on the interval \(-2 \pi, 2 \pi\). $$\csc x=\sqrt{2}$$

Short Answer

Expert verified
The solutions to the equation \(\csc x = \sqrt{2}\) in the interval (-2蟺, 2蟺) are 蟺/4, 3蟺/4, 5蟺/4, 7蟺/4, -蟺/4, -3蟺/4, -5蟺/4, and -7蟺/4.

Step by step solution

01

Determine the value of sin x

Since cosecant is the reciprocal function of the sine function, we first need to find when sin x equals 1/(\(\sqrt{2}\)). This occurs when x equals 蟺/4 and (3蟺)/4 in the first quadrant and when x equals (5蟺)/4 and (7蟺)/4 in the second quadrant.
02

Use Periodicity of Sine Function

The sine function (and hence, the cosecant function) has a period of 2蟺, which implies it repeats its values every 2蟺. Using this information, for every solution x found in Step 1, (x + 2n蟺), where 'n' is an integer, will also be a solution.
03

Apply given interval

The question asks for solutions within the interval (-2蟺, 2蟺). So the valid solutions within the given range are: 蟺/4, (3蟺)/4, (5蟺)/4, (7蟺)/4, -蟺/4, -3蟺/4, -5蟺/4, -7蟺/4.

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Key Concepts

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

Cosecant Function
The cosecant function, denoted as \(\csc x\), is the reciprocal of the sine function. This means \(\csc x = \frac{1}{\sin x}\). Since division by zero is undefined, the cosecant function is only valid where the sine function is not zero. This mainly includes all points except where \(x = k\pi\) (where \(k\) is an integer).
  • Key fact: \(\csc x = \sqrt{2}\) is equivalent to finding when \(\sin x = \frac{1}{\sqrt{2}}\).
  • The values for \(\sin x\) of \(\frac{1}{\sqrt{2}}\) are derived from the special angles \(\pi/4, 3\pi/4\).
Understanding the properties of the cosecant function, especially its connection to the sine function, is crucial for solving trigonometric equations.
Sine Function
The sine function, \(\sin x\), is a fundamental trigonometric function with several unique properties. It oscillates between -1 and 1, and its period is \(2\pi\), which means it repeats every \(2\pi\) radians.
  • The sine value of \(\frac{1}{\sqrt{2}}\) can be found at specific angles, like \(\pi/4\) or \(3\pi/4\) in the unit circle.
  • Understanding the unit circle is helpful: these angles appear symmetrically in different quadrants.
By knowing the angles where sine takes particular values, you can solve trigonometric equations involving the sine function more effectively.
Periodicity of Trigonometric Functions
Trigonometric functions, including sine and cosecant, have a periodic nature. This periodicity means that their values recur at regular intervals. For the sine function, this period is \(2\pi\).
  • If an equation holds for an angle \(x\), it also holds for \(x + 2n\pi\), where \(n\) is any integer.
  • This property allows us to extend solutions across multiple cycles of \(2\pi\) as seen in the exercise.
Grasping the concept of periodicity helps solve equations across different intervals by recognizing repeating patterns.
Graphical Solution of Equations
Graphical solutions entail plotting equations on a graph to visually identify solutions. For trigonometric equations, this can highlight the intervals where the solutions occur.
  • A graph of \(\csc x = \sqrt{2}\) would show intersections with a horizontal line at \(y = \sqrt{2}\).
  • These intersection points correspond to the solutions of the equation within the specified interval \((-2\pi, 2\pi)\).
Visualizing trigonometric equations can clarify their behavior and help identify solutions that may be less apparent algebraically.

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Most popular questions from this chapter

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{4} \sin 6 \pi t$$

During takeoff, an airplane's angle of ascent is \(18^{\circ}\) and its speed is 275 feet per second. (a) Find the plane's altitude after 1 minute. (b) How long will it take for the plane to climb to an altitude of 10,000 feet?

Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by \(y(t)=2 e^{-t} \cos 6 t,\) where \(y\) is the displacement (in centimeters) and \(t\) is the time (in seconds). Find the displacement when (a) \(t=0,\) (b) \(t=\frac{1}{4},\) and (c) \(t=\frac{1}{2}\)

Determine whether the statement is true or false. Justify your answer. $$\frac{\sin 60^{\circ}}{\sin 30^{\circ}}=\sin 2^{\circ}$$

Write the function in terms of the sine function by using the identity. $$A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right)$$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$f(t)=4 \cos \pi t+3 \sin \pi t$$
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