Phase shift and Applications - Revision history

Lila: /* Resources */

2021-10-26T20:18:58Z

Resources

Lila: /* Resources */

2021-10-08T17:36:36Z

Resources

Lila: /* Modeling Situations */

2021-10-08T17:34:17Z

Modeling Situations

Lila: /* Offset */

2021-10-08T17:33:32Z

Offset

Lila: /* Examples */

2021-10-08T17:32:12Z

Examples

Lila: /* Frequency */

2021-10-08T17:31:02Z

Frequency

Lila at 17:28, 8 October 2021

2021-10-08T17:28:49Z

Show changes

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:04:50Z

fixing link to go straight to pdf

Rylee.taylor: added Phase shift and Applications notes by professor Esparza

2020-08-11T21:15:02Z

added Phase shift and Applications notes by professor Esparza

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/ Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 20:18, 26 October 2021
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.
	* [https://en.wikibooks.org/wiki/Trigonometry/Phase_and_Frequency Phase and Frequencies], Wikibooks: Trigonometry/Calculus		* [https://en.wikibooks.org/wiki/Trigonometry/Phase_and_Frequency Phase and Frequencies], Wikibooks: Trigonometry/Calculus
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Trigonometry/Phase_and_Frequency Phase and Frequencies, Wikibooks: Trigonometry/Calculus] under a CC BY-SA license

← Older revision		Revision as of 17:36, 8 October 2021
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	==Resources==		==Resources==
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.
		+	* [https://en.wikibooks.org/wiki/Trigonometry/Phase_and_Frequency Phase and Frequencies], Wikibooks: Trigonometry/Calculus

@@ Line 161: / Line 161: @@
 The sinusoidal functions are very useful in real life applications, where these ''periodic'' (repeating) functions model situations that follow a pattern over a set amount of time, such as in modeling day length, planetary motion, sound waves, energy waves, etcetera.
-{{ExampleRobox|title=Fish length}}
+'''Fish length'''
 * Marine biologists determined the average length of <math>N=50,000</math> fish in a pond in millimeters (mm) is dependent on the time, in years. The table is shown below, where <math>t</math> is the time, in years, and <math>l(t)</math> is the average length of the fish, in millimeters. Find the sinusoidal function that best models the information provided.
@@ Line 202: / Line 202: @@
 There are many questions a person may be asking: where is the above circle in the question? Why use radians as opposed to degrees? . There are a few reasons, but it requires us to delve more deeply into mathematics that is usually considered advanced (polar coordinates). Unfortunately, Book 1 will not delve into this topic. However, for those that are curious, Book 3 will have polar coordinates that fully explain this behavior.
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.

@@ Line 105: / Line 105: @@
 Two sinusoidal waves may also have a different offset to each other, i.e. their average values are different, but they still have the same shape. Their graphs are y shifted relative to each other. In sound engineering this is called '''DC Bias'''.
-{{ExampleRobox|title=Different Offset}}
+'''Different Offset'''
 :<math>1+\cos(\theta)</math>
 has a different offset to <math>\cos(\theta)</math> . Whereas <math>\cos(\theta)</math> takes both positive and negative values, <math>1+\cos(\theta)</math> is never below 0.
 This offset is based off the '''vertical shift''', also known as the '''sinusoidal axis value'''. Keep in mind, unlike other functions, this vertical shift does <u>not</u> denote the y-intercept. The only way to determine the y-intercept correctly is to substitute zero into a function. Quick quiz: what is <math>\cos (0)</math>? In case you forgot, it is <math>1</math>. Therefore, the y-intercept for <math>f(\theta)=1+\cos(\theta)</math> is <math>f(0)=1+\cos(0)=1+1=2</math>.

@@ Line 83: / Line 83: @@
 ==Examples==
-{{ExampleRobox|title=Larger Amplitude}}
+'''Larger Amplitude'''
 :<math>10\cos(\theta)</math>
 has ten times the amplitude of <math>\displaystyle \cos \theta</math>
 The graph oscillates between 10 and -10 on the y axis.
+'''Larger Frequency'''
 :<math>\cos(100\theta)</math>
 has a higher frequency than <math>\cos(\theta)</math>
@@ Line 98: / Line 96: @@
 The wavelength is also decreased. The distance along the x axis for a complete cycle is one hundredth of what it was before.
-{{ExampleRobox|title=Different Phase}}
+'''Different Phase'''
 :<math>\sin(\theta)</math>
 has a different phase from <math>\cos(\theta)</math> . The phase difference happens to be 90°, because <math>\sin(\theta+90^\circ)=\cos(\theta)</math> . If two waves are 180° out of phase then where one is positive the other has the same value only negative. <math>-\sin(\theta)</math> is 180° out of phase with <math>\sin(\theta)</math>
 ==Offset==

@@ Line 3: / Line 3: @@
 Here are some sinusoidal (that is 'sine like') graphs. These show sinusoidal waves with different frequency. The frequency increases as you go down.
-{{ExerciseRobox|title=Waves and Light}}
+'''Waves and Light'''
 In physics, light is a kind of wave, and different frequencies of light give the different colors of the spectrum.
 *Does red light have a higher or lower frequency than blue light?
 *In a rainbow is red on the outside or the inside of the rainbow?  What about a double rainbow? (use the Internet)
 *Are radio waves higher or lower frequency than light waves? (use the Internet)
-{{ExerciseRobox|title=Waves and Sound}}
+'''Waves and Sound'''
 In physics sound is a kind of wave. It's physically a different kind of wave from light, but its vibrations can still be described using sines and cosines. Light is an electromagnetic wave and sound is a wave involving change in air pressure. That difference is the underlying reason why light can travel through the vacuum of space, but sound cannot.
@@ Line 17: / Line 16: @@
 *Explain in your own words what 'Hertz' means, and what the K in the abbreviation KHz means.
 *What vibration frequencies are typical in an Earthquake?
 ==Sine and Cosine Graphs==

← Older revision		Revision as of 22:04, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/ Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Phase%20shift%20and%20Applications/Esparza%201093%20Notes%202.6.pdf Phase shift and Applications]. Written notes created by Professor Esparza, UTSA.