# What is Sinusoidal function: Definition and 21 Discussions

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

where:

A, amplitude, the peak deviation of the function from zero.
f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second

φ

{\displaystyle \varphi }
, phase, specifies (in radians) where in its cycle the oscillation is at t = 0. When

φ

{\displaystyle \varphi }
is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

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1. ### A Calculate the exact or approximate length of one period of the wave function

The known expression of the wave function is where A is the amplitude, k the wave number and ω the angular velocity. The mathematical definition of arc length for a generical function in an interval [a,b] is where, in our sinusoidal case: For our purpose (calculation of the length in one...
2. ### A Spatial Fourier Transform: Bessel x Sinusoidal

I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
3. ### Derivative of a sinusoidal function

Homework Statement What is the derivative of ##f(x)=\frac {2x^2} {cos x}##? Homework EquationsThe Attempt at a Solution ##F(x)=\frac {2x^2} {cos x}## So... ##f(x)=2x^2## and ##f'(x)=4x## ##g(x)=cosx## and ##g'(x)=-sinx## If I plug these into the quotient rule I thought that I would get...
4. ### Show that the function is a sinusoid by rewriting it

Homework Statement Hello! I am doing exercises on sinusoid functions from the beginning of Trigonometry. I hoped I understood the topic, but it seems not quite, because I don't get the results authors show as examples for one of possible answers, as there can be a few answers to the same...
5. ### A Finding the frequency of sinuosid in a constant + sinusoid?

Hi everyone, I am working on some problem relating to radars. The problem boils down to finding the frequency of the complex exponential in a constant + complex exponential + noise model. I found some papers on sinusoid recognition but they use the sinusoid + noise model only. I tried to come...
6. ### Exercise about the wavefunction

Homework Statement [/B] Consider an ideal rope where there is a wave moving at velocity ##v=20 m/s##. The displacement of one end of the rope is given by $$s(t)=0.1 \mathrm{sin}(6 t)$$ a) Find the wavefunction ##\xi(x,t)##, knowing that it is progressive b) Find the distance ##\delta## (in...
7. ### Equation for Simple Harmonic Motion?

In Simple Harmonic Motion, can (k/m) = ω2 be expressed for all SHMs or only the ones in which the mass due to which the SHM is being executed is performing a circular motion? Since for example, in the case of spring, there is no circular motion involved, so omega should not be defined for...
8. ### Finding the Intersection of a Sinusoidal Function and a Line

Homework Statement Hi! I'm trying to find the points of intersection of a sinusoidal function and a line. The line is y=x/7. The function is y=sinx. Can someone tell me how to determine the number of intersections and exact intersections. I would also like to know if the same method can be...
9. ### MHB How do I find A,B,C, and D in a sinusoidal function?

I really need someone to break it down for me. I think I understand A and D, but I am confused on B and C. I have some example problems. But first, the equation my pre-calculus teacher has given us is y=Asin(2π/B(θ-C))+D. But I am still having a lot of trouble. Find amplitude, period, a phase...
10. ### Sinusoidal function. Top percentage of values.

Homework Statement For a sinusoidal function, how do you determine the highest value exceeded 10% of the time? The pink line in the attached pic indicates that value. Just wondering how you actually determine the value for a periodic function? Homework Equations The Attempt...
11. ### MHB Sarah's question at Yahoo Answers regarding a sinusoidal function

Here is the question: Here is a link to the question: One of the largest ferris wheel ever built is in the british airways london eye which was completed in 2000. T? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
12. ### Find derivative of complex sinusoidal function

Homework Statement Find derivative of: h(x) = 3e^{sin(x+2)} Homework Equations chain rule of derivatives, product rule(?) The Attempt at a Solution I'm quite sure I'm doing this wrong. Because the exponent is a product, for the derivative of the exponent I would have to use the...
13. ### Derivative of a sinusoidal function. Need some Expertise

Hey forum, thanks in advance to anyone who can explain this to me! Homework Statement Find the derivative of each of the following: h(x) = 3e^sin(x + 2) The Attempt at a Solution My friend and I worked through this together and solved for: h’(x) = (3e^sin(x + 2))(cos(x + 2)) Using...
14. ### Sinusoidal function, find its parameters

Homework Statement Measurements of the height h(t) of water in a harbor are recorded ,where h is measured in meters and t in hours.It was noted that the rise and fall of a tide is modeled by a sinusoidal function giving the height by : h(t)=a+bsin(kt+c). (a) Obtain values of the...
15. ### Equation of the sinusoidal function that represents height above the ground

Question: 1. A clock is hanging on wall. length of minute hand is 16cm and the length is 8cm for the hour hand. The highest that the tip of the minute hand reaches above ground is 265cm. a) What is equation of axis, amplitude & period in minutes of function that represents the tip of hour...
16. ### Find the equation of the tangent to the curve (sinusoidal function)

Homework Statement Find the equation of the tangent to the curve y=2cos^3x at x=pi/3 Homework Equations The Attempt at a Solution y=2cos^3x dy/dx=-6sinxcos^2x 0=-6sinxcos^2x set x = pi/3 and solve for the derivative, plug the answer into y=2cos^3x where do I go from here...
17. ### How to represent light as a sinusoidal function

I have seen many equations light as a wave but none of which have represented light as a simple sinusoidal function. I want to be able to graph light waves on a cartesian plane as one sine curve. I know that light isn't that simple but could light be represented as a 2 dimensional sine curve...
18. ### Applying a sinusoidal function to tidal patterns on the earth.

Homework Statement At the time of a full moon, the tides repeat with a period of about 12 hours, and the depth of water in a certain channel varies between 2 meters and 6 meters in a way that can be modeled by an equation of the form D(t) = A + Bsin(ct + d), where A is the average depth and...
19. ### Metal Heat Treatment: How to create a sinusoidal function that increases in frequency

Alright, this is a hypothetical problem for my math class, (but it seemed to fit here better than calculus help) and though I am allowed to site sources, I don't necessarily want the answer just given to me (I mean, I'd really like to figure it out). Homework Statement Treatment lasts...
20. ### Exploring the Effect of Large δ on Sinusoidal Function at x=1.5

In a sinusoidal function...suppose the value of δ is very large...then as x approaches any a, the value of f(x) might not approach L directly...or there should not be a direct relation; example - \lim_{x \to 1.5} sin x = 0.997494986 Where I've stated δ as 7...then if x = 1.5 – 6.9 =...
21. ### Solving Sin(2x)+Sin(x)=0 over [0,2π)

hello there, how do i solve over [0,2pi) for sin(2x)+sin(x)=0 thanks for what help you can give RansidMeat