Unit 2 power point
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This document discusses the key components of periodic wave functions, including amplitude, period, phase shift, and their roles in trigonometric functions like sine and cosine waves. It provides examples of how changing the amplitude, period, and phase shift in an equation like y = Asin(Bx + C) + D impacts the resulting wave graph. The author indicates understanding that these components are needed to determine the type of periodic wave from an equation and that there are formulas to calculate the period, amplitude, and phase shift.
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Unit 2 power point
The document discusses the functions f(t)=A sin (Bt + C) and f(t)=A cos (Bt + C), explaining that they require an amplitude (A), period (determined by B), and phase shift (C) to define the wave pattern. It provides the formulas for these components and indicates that the amplitude is the distance from the midpoint to the highest/lowest points, while the phase shift represents horizontal displacement from the original position. An example function is given with defined values for amplitude, period, and phase shift.
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This document discusses periodic graphs and phase shifts of trigonometric functions. It defines the standard forms for sine and cosine functions and explains that:
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It provides examples of identifying amplitude, period, phase shift, and vertical shift from trigonometric equations. It explains that phase shift results in horizontal translation of the graph while vertical shift results in vertical translation.
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Periodic functions repeat exactly in regular intervals called cycles. The trigonometric functions sine and cosine are periodic functions that can be used to model real-world phenomena like sound waves. Sine and cosine functions can be modified through transformations including amplitude, period, vertical shift, and phase shift. These transformations allow sine and cosine graphs to model diverse physical situations.
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The document discusses graphing tangent, cotangent, secant, and cosecant functions. It provides examples of how to identify the period, x-intercepts, and asymptotes in order to graph transformations of these trigonometric functions. Key steps include using the graphs of sine, cosine, or tangent as a guide and determining the period from the transformation parameter before identifying intercepts and asymptotes based on the period. Examples are worked through and a quiz is provided to assess understanding.
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- Steps for solving trigonometric equations by taking the inverse sine or cosine of both sides.
Chap 1 trigonometry 2 part 1
The document discusses trigonometric graphs and their key characteristics. It provides examples of trigonometric functions and shows how to determine the period, amplitude, and phase shift of sine and cosine graphs. The period is 2π/b, the amplitude is the absolute value of a, and the phase shift is the absolute value of c, which shifts the graph right or left depending on whether c is positive or negative. Examples are worked through step-by-step to illustrate how to find these values and sketch the graph of one period.
Graphing day 1 worked
This document discusses graphing sine and cosine functions. It defines periodic functions and notes that sine and cosine have a period of 2π. The properties of sine are described as having a domain of (-∞, ∞), range from -1 to 1, and amplitude equal to the absolute value of A. Similarly, cosine has a period of 2π, domain of (-∞, ∞) and range from -1 to 1. Translating sine and cosine functions is also discussed, noting that amplitude and period can change based on coefficients of x and constants added. Examples are provided to demonstrate stating the amplitude and period of various sine and cosine graphs.
Circular (trigonometric) applications
1. The document provides revision on circular functions and common student errors. It discusses when to use radian or degree mode and how to convert between the two.
2. It reviews exact trigonometric values, the CAST circle, graph properties of sin, cos and tan, and solving trigonometric equations.
3. Two example problems are given, one modeling heart rate with sine and another modeling bungee jumping height with cosine. Key values are determined from the graphs like initial height, minimums, and period.
Trigonometric functions
This document discusses trigonometric functions and their properties. It defines periodic functions and their periods. It describes the amplitude of sine and cosine functions as half the difference between the maximum and minimum values. It discusses how transformations of a, b, h, and k values can stretch, compress, reflect, translate and shift the graphs of sine and cosine functions and how these affect their amplitudes and periods. Examples are provided to demonstrate identifying amplitudes, periods, phase shifts from transformed trigonometric functions.
4. a Find the compact trigonometric Fourier series for the periodic s.pdf
4. a Find the compact trigonometric Fourier series for the periodic signal x() and sketch the
amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra,
sketch the exponential Fourier spectra. for x(t) c) By inspection of spectra in part b), write the
exponential Fourier series (20 points) X(t) T/2 a/2 T/2
Solution
The Fourier series is a method of expressing most periodic, time-domain functions in the
frequency domain. The frequency domain representation appears graphically as a series of spikes
occurring at the fundamental frequency (determined by the period of the original function) and
its harmonics. The magnitudes of these spikes are the Fourier coefficients. This series of
components are called the wave spectrum. THE EXPONENTIAL FOURIER SERIES å ¥ =-¥ w
= n jn t n f t D e 0 ( ) where: D is the amplitude or coefficient n is the harmonic w0 is the radian
frequency [rad/s] w0/2p is the frequency [hertz] T0 = 2p/w0 is the period [sec.] ò - w = 0 0 ( ) 1
0 T jn t n f t e dt T D 0 0 0 D = C = a 0 0 2 T p w = The exponential form carries no more and no
less information than the other forms but is preferred because it requires less calculation to
determine and also involves simpler calculations in its use. One drawback is that the exponential
form is not as easy to visualize as trigonometric forms. THE TRIGONOMETRIC FOURIER
SERIES å ¥ = = + w + w 1 0 0 0 ( ) cos sin n n n f t a a n t b n t where: a is the amplitude or
coefficient n is the harmonic q is the phase w0 is the radian frequency [rad/s] w0/2p is the
frequency [hertz] T0 = 2p/w0 is the period [sec.] ò = 0 ( ) 1 0 0 T f t dt T a 0 0 2 T p w = n n T n
f t n t dt C T a = w = q ò ( ) cos cos 2 0 0 0 n n T n f t n t dt C T b = w = - q ò ( )sin sin 2 0 0 0
The Fourier series of an even periodic function will consist of cosine terms only and an odd
periodic function will consist of sine terms only.
Tom Penick tomzap@eden.com www.teicontrols.com/notes 2/5/2000 THE COMPACT
TRIGONOMETRIC FOURIER SERIES å ¥ = = + w + q 1 0 0 ( ) cos( ) n n n f t C C n t where:
C is the amplitude or coefficient n is the harmonic q is the phase w0 is the radian frequency
[rad/s] w0/2p is the frequency [hertz] T0 = 2p/w0 is the period [sec.] 2 2 Cn = an + bn 0 0 C = a
n n a , b : see above ÷ ÷ ø ö ç ç è æ - q = - n n n a 1 b tan 0 0 2 T p w = THE DIRICHLET
CONDITIONS WEAK DIRICHLET CONDITION - For the Fourier series to exist, the function
f(t) must be absolutely integrable over one period so that coefficients a0, an, and bn are finite.
This guarantees the existence of a Fourier series but the series may not converge at every point.
STRONG DIRICHLET CONDITIONS - For a convergent Fourier series, we must meet the
weak Dirichlet condition and f(t) must have only a finite number of maxima and minima in one
period. It is permissible to have a finite number of finite discontinuities in one period.
FREQUENCY SPECTRA AMPLITUDE SPECTRUM - The plot of amplitude Cn versus the
(radian) frequency.
MA101-Lecturenotes(2019-20)-Module 13 (1).pdf
This document defines scalar and vector fields and provides examples of each. It also discusses line integrals, which integrate a scalar or vector field over a curve. Line integrals of vector fields can be used to calculate work done by a force field or fluid flow along a curve. The line integral is path independent for conservative vector fields, which can be expressed as the gradient of a potential function.
MA101-Lecturenotes(2019-20)-Module 13 (1).pdf
- A scalar field assigns a scalar value to each point in a region of space, like temperature. A vector field assigns a vector, like fluid velocity.
- Line integrals calculate the work done or fluid flow along a path by integrating the dot product of the field and the path's tangent vector.
- Conservative vector fields have a potential function whose derivatives give the field. Line integrals of conservative fields depend only on the endpoints and not the path taken.
Learning object 1
The document discusses phase and phase difference in waves. It defines phase as being dependent on position and time, represented by the Greek symbol φ in the equation D(x,t) = Asin(kx±ωt+φ). Phase difference is the difference in phase between two positions at the same time. Points an integer number of wavelengths apart are in phase, while half-integer wavelengths apart are out of phase. The equation for phase difference is derived as ∆Φ=k(x2-x1).
Math Project !.docx What is the effect of changing the transformations on a s...
1) Changing the amplitude multiplies the maximum and minimum values of the sine wave vertically. Doubling amplitude stretches the wave vertically while halving amplitude compresses it.
2) Changing the period stretches or compresses the wave horizontally. Halving period increases it while doubling decreases it.
3) Changing the phase shift moves the wave left or right along the x-axis. Increasing phase shift moves it left and decreasing moves it right.
4) Changing the vertical shift moves the entire wave up or down along the y-axis. Increasing shift moves it up and decreasing moves it down.
Differentiation using First Principle - By Mohd Noor Abdul Hamid
1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.
pRO
The document discusses modeling a flood in a river using 1D Saint-Venant equations. It presents the mass and momentum equations, describes common routing methods varying in complexity, and provides details on explicit and implicit numerical schemes to solve the equations. It also explains the HEC-1 hydrologic routing method and provides sample input parameters and results from a test model run.
Amplitude and Period
This document discusses how to mathematically modify properties of a sine wave sound model. It explains that amplitude determines volume and is represented by the constant multiplied to the sine function. Period determines tone and is the time for one full wave cycle, which is related to the constant inside the sine. Doubling the inside constant halves the period, speeding up the waves. Manipulating these constants allows controlling the sound wave properties mathematically.
Trigonometry
The document summarizes trigonometric functions and their relationships. It discusses the unit circle, radians vs degrees, computing trig ratios using the unit circle, trig identities, trig functions and their properties including amplitude, frequency, phase and period. It provides examples of trig functions and applications to modeling periodic behavior like sound waves and tides. Practice problems at the end involve converting between radians and degrees and expressing trig ratios as radicals.
Annotations 2
The document discusses modeling tidal data with a sine wave function. It provides steps to sketch the graph based on key components:
- D is the vertical shift (average sea level of 0 feet)
- A is the amplitude of 2 feet
- B determines the period of 12.5 hours
- C is the horizontal shift starting at January 1st
It describes finding the equation to model the tide heights over time by determining the A, B, C, and D values. The example uses the equation y=2sin((π/12.5)x) , solves for theta, and inputs back into the equation to find the first high tide was at 1:47 am on January 1st.
03 Cap 2 - fourier-analysis-2015.pdf
This document discusses Fourier analysis and periodic functions. It defines periodic functions and introduces Fourier series representation of periodic functions as the sum of sinusoidal components. The key points are:
1) A periodic function can be represented by an infinite series of sinusoids of harmonically related frequencies, known as Fourier series representation.
2) The coefficients of the Fourier series (a0, an, bn) can be calculated from the integrals of the function over one period.
3) Periodic functions may exhibit odd symmetry, even symmetry, or half-wave symmetry, which determines which coefficients (an or bn) are zero.
4) Examples demonstrate the Fourier analysis of specific periodic waveforms, such as a
Similar to Unit 2 power point (20)
4. a Find the compact trigonometric Fourier series for the periodic s.pdf
4. a Find the compact trigonometric Fourier series for the periodic s.pdf
Math Project !.docx What is the effect of changing the transformations on a s...
Math Project !.docx What is the effect of changing the transformations on a s...
Differentiation using First Principle - By Mohd Noor Abdul Hamid
Differentiation using First Principle - By Mohd Noor Abdul Hamid
Unit 2 power point
- 1. By : Valeria Sanchez
- 2. Graph |a| = amplitude 2pi/b = period c/b = phase shift The amplitude (A) is the distance from the midpoint to the highest or lowest point of the function. Phase shift is the amount of horizontal displacement of the function from its original position. The period (T) is the distance between any two repeating points on the function Sin function Cosine function y = Asin(Bx + C) + D y = Acos(Bx + C) + D Functions of the form f(t)=A sin (Bt + C) & f(t)=A cos (Bt + C)
- 3. EXAMPLE: amplitude 'A' = 1.5, period 'T' = p/2, phase shift = p/3 and 'y' shift = 0.5 b=4 c=4/3 π Graph: y = 1.5sin (4x + 4/3π) + 0.5 My understanding of unit 2 is that we need an amplitude, period, and phase shift to determine what kind of periodic wave is going to come out of the equation. I have also learned that there’s a formula to finding out the period, amplitude, and phase shift. I am also aware that if its a positive then the periodic wave is going to go to the left and that if its a negative then it goes to the right.