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Chapter 5: Problem 4

State the amplitude of each periodic function. Sketch the graph of each function. a) \(y=2 \sin \theta\) b) \(y=\frac{1}{2} \cos \theta\) c) \(y=-\frac{1}{3} \sin x\) d) \(y=-6 \cos x\)

Short Answer

Expert verified
a) Amplitude = 2 b) Amplitude = \frac{1}{2}\ c) Amplitude = \frac{1}{3}\ d) Amplitude = 6

Step by step solution

01

- Identify the amplitude

The amplitude of a periodic function of the form \(y = A \sin \theta\) or \(y = A \cos \theta\) is the absolute value of the coefficient A. This tells us the maximum distance from the midline to the peak of the wave.
02

- Amplitude of each function

a) For \(y = 2 \sin \theta\), the coefficient A is 2. Thus, the amplitude is \[ |2| = 2 \] b) For \(y = \frac{1}{2} \cos \theta\), the coefficient A is \frac{1}{2}\. Thus, the amplitude is \[ |\frac{1}{2}| = \frac{1}{2} \] c) For \(y = -\frac{1}{3} \sin x\), the coefficient A is -\frac{1}{3}\. Thus, the amplitude is \[ |\frac{1}{3}| = \frac{1}{3} \] d) For \(y = -6 \cos x\), the coefficient A is -6. Thus, the amplitude is \[ |6| = 6 \]
03

- Sketch each graph

To sketch the graph: \ 1. For \(y = 2 \sin \theta\), draw a sine wave with peaks at 2 and troughs at -2. 2. For \(y = \frac{1}{2} \cos \theta\), draw a cosine wave with peaks at \frac{1}{2}\ and troughs at -\frac{1}{2}\. 3. For \(y = -\frac{1}{3} \sin x\), draw a sine wave reflected over the x-axis with peaks at -\frac{1}{3}\ and troughs at \frac{1}{3}\. 4. For \(y = -6 \cos x\), draw a cosine wave reflected over the x-axis with peaks at -6 and troughs at 6.

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Key Concepts

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

Sin Function
The sine function, represented as \(y = \sin \theta\), is a fundamental periodic function. It's widely used in trigonometry and represented by a wave-like pattern. The sine function oscillates between -1 and 1. One full cycle of \(\sin \theta\) occurs over an interval of \[0, 2\pi \] on the x-axis. Key points to note:
  • At \theta = 0\, the function starts at 0.
  • It reaches its maximum value of 1 at \theta = \frac{\text{\pi}}{2}\.
  • It returns to 0 at \theta = \pi\.
  • It hits its minimum value of -1 at \theta = \frac{3\text{\pi}}{2}\.
  • Completes the cycle back at 0 when \theta = 2\pi\.
This pattern continues indefinitely in both directions. You will observe similar characteristics with transformed sine functions like \(y = A \sin \theta\), where the coefficient \(A\) affects the amplitude.
Cos Function
The cosine function, denoted as \(y = \cos \theta\), is another core periodic function in trigonometry. It has a similar wave-like appearance but starts at a maximum value. The cosine function also moves between -1 and 1. Over an interval of \[0, 2\pi \], the key points are:
  • At \theta = 0\, the function begins at 1.
  • It goes down to 0 at \theta = \frac{\text{\pi}}{2}\.
  • It reaches -1 at \theta = \pi\.
  • It moves back up to 0 at \theta = \frac{3\text{\pi}}{2}\.
  • Finally, it returns to 1 at \theta = 2\pi\.
Like the sine function, the cosine function can also be transformed. With \(y = A \cos \theta\), the coefficient \(A\) determines the amplitude, stretching or compressing the wave vertically.
Graphing Periodic Functions
To graph periodic functions like \(y = \sin \theta\) or \(y = \cos \theta\), follow these steps:
  • Identify the amplitude from the coefficient \(A\). This defines the maximum value the graph reaches from its midline.
  • Determine the period of the function. For \(\sin \theta\) and \(\cos \theta\), the period is \2\pi\.
  • Find key points, such as maximums, minimums, and points where the function crosses the midline (usually at 0).
  • Plot these points accurately on your graph.
  • Connect the points smoothly to form the characteristic wave shape.
  • If the coefficient is negative, reflect the graph over the x-axis. For example, with \(y= -\cos \theta\), peaks become troughs and vice versa.
Practicing by sketching multiple periodic functions helps reinforce understanding and aids in recognizing their patterns.
Absolute Value of Coefficient
The absolute value of a coefficient in functions like \(y = A \sin \theta\) or \(y = A \cos \theta\) is critical for determining the amplitude. Amplitude is the height of the wave from the midline to the peak. Here's how to find it:
  • Take the coefficient \(A\) in front of \sin\ or \cos\.
  • Calculate its absolute value, represented as \|A|\. This removes any negative sign, ensuring the amplitude is positive.
For example:
  • In \(y = 2 \sin \theta\), the amplitude is \|2| = 2\.
  • In \(y = -6 \cos x\), the amplitude is \|6| = 6\.
Understanding this concept allows you to accurately plot the maximum and minimum points on a graph, enhancing your ability to graph these functions effectively.

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Most popular questions from this chapter

Determine the phase shift and the vertical displacement with respect to \(y=\sin x\) for each function. Sketch a graph of each function. a) \(y=\sin \left(x-50^{\circ}\right)+3\) b) \(y=\sin (x+\pi)\) c) \(y=\sin \left(x+\frac{2 \pi}{3}\right)+5\) d) \(y=2 \sin \left(x+50^{\circ}\right)-10\) e) \(y=-3 \sin \left(6 x+30^{\circ}\right)-3\) f) \(y=3 \sin \frac{1}{2}\left(x-\frac{\pi}{4}\right)-10\)

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)

If \(\sin \theta=0.3,\) determine the value of \(\sin \theta+\sin (\theta+2 \pi)+\sin (\theta+4 \pi)\)

A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. When the mass is released, it takes \(0.3 \mathrm{s}\) to reach a high point of 60 cm above the floor. It takes 1.8 s for the mass to reach the first low point of \(40 \mathrm{cm}\) above the floor. a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads \(17.2 \mathrm{s?}\) d) What is the first positive value of time when the mass is \(59 \mathrm{cm}\) above the floor?

After exercising for 5 min, a person has a respiratory cycle for which the rate of air flow, \(r,\) in litres per second, in the lungs is approximated by \(r=1.75 \sin \frac{\pi}{2} t,\) where \(t\) is the time, in seconds. a) Determine the time for one full respiratory cycle. b) Determine the number of cycles per minute. c) Sketch the graph of the rate of air flow function. d) Determine the rate of air flow at a time of 30 s. Interpret this answer in the context of the respiratory cycle. e) Determine the rate of air flow at a time of 7.5 s. Interpret this answer in the context of the respiratory cycle.
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