Chapter 15: Problem 7
Graph each sine wave. Find the amplitude, period, and phase shift.$$y=\sin \left(x+15^{\circ}\right)$$
Short Answer
Expert verified
The amplitude is 1, the period is \(360^\circ\) or \(2\pi\) radians, and the phase shift is \(-15^\circ\) to the left.
Step by step solution
01
Determine the Amplitude
The amplitude of a sine wave is the coefficient of the sine function. In this case, the sine function is given as \(y = \sin(x + 15^\circ)\), and since there is no coefficient in front of the sine function, the amplitude is 1.
02
Identify the Period
The period of the standard sine function, \(y = \sin(x)\), is \(2\pi\) radians or \(360^\circ\). Since the function has no horizontal compression or expansion, \(y = \sin(x + 15^\circ)\) also has a period of \(2\pi\) radians or \(360^\circ\).
03
Find the Phase Shift
The phase shift of the sine function can be determined from the horizontal shift inside the trigonometric function. In \(y = \sin(x + 15^\circ)\), the phase shift is \(-15^\circ\), meaning the graph shifts 15 degrees to the left.
04
Graph the Sine Wave
Begin by plotting the key points of the sine wave starting with the phase shift. Plot a sine wave with an amplitude of 1, a period of \(360^\circ\), and a phase shift of \(-15^\circ\). The sine wave will reach its maximum at 1 and minimum at -1, with zero crossings at \(-15^\circ\), \(165^\circ\), \(345^\circ\), etc., and maxima and minima at \(75^\circ\), \(255^\circ\), etc.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude of Sine Wave
When graphing a sine wave, one of the primary features to identify is the amplitude. The amplitude of a sine wave is a crucial concept in trigonometry, as it represents the wave's maximum distance from its central axis. Think of it as the height of the wave's crest or the depth of its trough from the equilibrium position.
In the example, the function given is \(y = \sin(x + 15^\circ)\), and there is no number multiplying the sine function. This absence indicates that the amplitude is the default value of 1. It's worth noting that had there been a coefficient, say \(a\), the amplitude would be the absolute value of \(a\). As such, the graph's peaks would reach \(a\) units above the axis, and the troughs would dip \(a\) units below.
For students to fully grasp the amplitude's impact on the wave, it's helpful to visualize how the wave would look if the amplitude were different. For instance, an amplitude of 2 would make the wave appear taller, with peaks and valleys twice as high or low as the original.
In the example, the function given is \(y = \sin(x + 15^\circ)\), and there is no number multiplying the sine function. This absence indicates that the amplitude is the default value of 1. It's worth noting that had there been a coefficient, say \(a\), the amplitude would be the absolute value of \(a\). As such, the graph's peaks would reach \(a\) units above the axis, and the troughs would dip \(a\) units below.
For students to fully grasp the amplitude's impact on the wave, it's helpful to visualize how the wave would look if the amplitude were different. For instance, an amplitude of 2 would make the wave appear taller, with peaks and valleys twice as high or low as the original.
Period of Sine Function
The period of a sine function defines the length of one complete cycle of the wave. In simpler terms, it's the distance on the x-axis that the wave takes to repeat its shape. The standard sine function, \(y = \sin(x)\), makes this full cycle every \(2\pi\) radians or \(360^\circ\), which we recognize as one complete revolution around the unit circle in trigonometry.
When students graph the function, they should measure one period along the x-axis to identify where the pattern will start repeating. Understanding this concept ensures that they can accurately graph waves of different frequencies and correctly interpret real-world applications like sound waves or alternating current electricity.
Calculating the Period
If the sine function had been modified, such as \(y = \sin(bx)\), the period would be \(\frac{2\pi}{|b|}\) or \(\frac{360^\circ}{|b|}\). However, in our exercise \(y = \sin(x + 15^\circ)\), there is no coefficient \(b\), thus the period remains unchanged at \(2\pi\) radians or \(360^\circ\).When students graph the function, they should measure one period along the x-axis to identify where the pattern will start repeating. Understanding this concept ensures that they can accurately graph waves of different frequencies and correctly interpret real-world applications like sound waves or alternating current electricity.
Phase Shift in Trigonometry
The phase shift in trigonometry is the horizontal displacement of the sine (or cosine) wave from its standard position. It indicates how far and in which direction the wave is shifted along the x-axis. A positive phase shift moves the graph to the right, while a negative one moves it to the left.
In the exercise, we work with the function \(y = \sin(x + 15^\circ)\). The addition of \(15^\circ\) inside the sine function suggests a horizontal shift. Since it's \(x + 15^\circ\), the shift is \( -15^\circ\), moving the graph 15 degrees to the left from its standard position. To visualize, students should think about starting their graph 15 degrees before the usual starting point of a sine wave.
In the exercise, we work with the function \(y = \sin(x + 15^\circ)\). The addition of \(15^\circ\) inside the sine function suggests a horizontal shift. Since it's \(x + 15^\circ\), the shift is \( -15^\circ\), moving the graph 15 degrees to the left from its standard position. To visualize, students should think about starting their graph 15 degrees before the usual starting point of a sine wave.