What is Cosine: Definition and 342 Discussions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

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  1. R

    Question about cosine and Eulers identity

    I was doing a signals and systems problem and I think I might be screwing something up with the cosine function because I get cos(a+b) = cos(a)*cos(b) This is how cos(a+b)=Re\left\{ e^{j*(a+b)} \right\} =Re\left\{ e^{j*(a)}*e^{j*(b)} \right\} =cos(a)*cos(b) Can anyone point out my mistake...
  2. D

    Newton's second law cosine and sine

    Newton's second law is taking my mind for a spin and for some reason had me contemplating how it works for several hours. This is all with respect to an incline and an object sitting on the incline with no friction. If the problem does not give you the mass of the object can you completely...
  3. D

    Fourier Series Representation of a Square Wave using only cosine terms.

    Hello, I am attempting a past exam paper in preparation for an upcoming exam. The past exam papers do not come with answers and I'm a little unsure as to whether I'm doing all of the questions correctly and would like some feedback if I'm going wrong somewhere. Any help is greatly appreciated...
  4. S

    Evaluating a Limit with Cosine Functions

    Evaluate the following limit: \lim_{n\rightarrow \infty}\frac{1+\cos(\frac{x}{n})+...+\cos(\frac{n-1}{n}x)}{n}
  5. H

    Orthogonality of Sine and Cosine functions

    Hi, would anyone be able to explain how to evaluate a function using orthogonality (i.e. using orthogonality to solve a definite integration problem with sines/cosines)? Thank you
  6. M

    Understanding the Transition to Solve for t in an Oscillation Function

    I have a cosine function, namely the function for oscillation x=A cos(wt). I want to separate the t out here, so I can solve for it. My teacher gave the the answer to be t= (arccos (x/a)/2pi)*T, but I can't quite see where he came up with that. Would anyone be as kind as to give me a more...
  7. A

    Fourier cosine series of cos(x) from x=0 to Pi

    Homework Statement Find the Fourier cosine series for f(x) = cos x, 0 < x < Pi EDIT: I believe we are talking about a half-range extension.. Homework Equations Fourier cosine series: f(x) = a0/2 + Sum(n = 1 to infinity) (an * cos (nx)) where a0 = 2/L * integral (x = 0, x = L)...
  8. D

    Challenging problem corrrection no sine and no cosine law

    sorry about the mistake in my last post. I miswrote the bottom vertex of the equilateral triangle. Let me re-state the problem correctly This is the 3rd and final question I post from the book, The Unsolvable and the Solvable. It is NOT a homework question. This is something for...
  9. D

    Another challenging question, no sine and no cosine law

    This is the 3rd problem from the book, The Unsolvable and the Solvable. This is the last one I post from this book. It is pretty challenging. Once again, it is NOT a homework problem. Consider an isoceles triangle ABC and an equilateral triangle BCD which share the side BC as shown...
  10. D

    Exploring the Origins of Sine and Cosine Formulas

    Hello. I know the definition of sine and cosine, but how were these formulas originally invented? I mean, how did people derive the power series for sine and cosine for the first time?
  11. D

    Complex analysis definite integral involving cosine

    Homework Statement integral 1/(a+cos(t))^2 from 0 to pi. Homework Equations cos(t)=1/2(e^it+e^-it) z=e^it dz/(ie^it)=dt The Attempt at a Solution int dt/(a+cos(t))^2 = int dz/iz(a2+az+az-1+z2/4 +1/2 +z-2/4) so with these types of problems I normally can factor this guy...
  12. A

    Proof of Cosine law using vectors

    Homework Statement Two vectors of lengths a and b make an angle θ with each other when placed tail to tail. Prove, by taking components along two perpendicular axes, that the length of the resultant vector is r=√(a^2+b^2+2abcos θ ) Homework Equations Would this be correct? How would you...
  13. R

    Point of intersection for sine and cosine functions

    The problem is finding the points of intersection for two given functions. f1=sin(-\pi*x) f2=1+cos(-\pi*x) I've plotted the functions using Maple. http://dl.getdropbox.com/u/12485/plot.png And I'm quite certain that to find the points of intersection, I have to set f1=f2 which...
  14. S

    Solving an Equation with Cosine

    1.x = rcos(a)cos(a) x is known r is known I can't seem to get this! x = -2 r = 7 therefore -2/7 = cos(a)*cos(a) so.. -2/7 = cos^2(a) i know cos^2(a) = 1 + cos2a / 2 -2/7 = 1 + cos2a / 2 so -4/7 -1 = cos2a I cannot take cos inverse of this value since it is...
  15. T

    The sum of series involving cosine

    Homework Statement Find the sum of the series s(x) = 1 +cos(x)+ (cos2x)/2!+(cos3x)/3!... Homework Equations The Attempt at a Solution
  16. thenewmans

    How is cosine squared used in measuring particle properties?

    I have a few questions: What’s that formula called? The correlation formula? Is cos^2 the formula for only particles with spin=1 like photons? If so, what’s used for electrons (1/2 spin). Also, cos^2 seams funny to me since it returns 1 for 0 and 180 degrees. Zero I can understand since you...
  17. J

    Lambert's Cosine Law: Understanding Diffuse Reflection

    Hello, I have a quick question relating to Lambert's cosine law for diffuse reflection. My understanding of the law is that given an isotropically illuminated surface, the reflection distribution goes with the cosine of the zenith angle. Now my question is whether or not that law holds for...
  18. S

    Linearizing Cosine: Finding Derivatives w/Moivre's

    Homework Statement I make the product \cos(x)\cos(5x) a sum of two cosines. I think this is called linearizing but please correct me if I'm wrong. (I need it to find the n-th derivative of that function). Homework Equations The Attempt at a Solution I know the answer is...
  19. M

    Fourier Cosine Transform and Complex Exponential Solution for Homework Problem

    Homework Statement given 2 functions f and g related by a cosine transform g( \alpha ) = \int_{0}^{\infty}dx f(x)Cos( \alpha x) then if the integral \int_{0}^{\infty}dx f(x)exp(cx) exists for every positive or negative 'c' then should it be equal to \int_{0}^{\infty}dx...
  20. V

    Fourier Transform of cosine and rect

    Homework Statement Just wanted to check if I did the Fourier transform of a somewhat long function correctly Homework Equations f(x) = (1+cos(\frac{2pix}{w}))rect2(\frac{x}{w}) they're not convolutions, just a modulation equation used in imaging studies 'rect' is rectangle function...
  21. M

    Exploring Inequalities of Cosine: Understanding the Relationship Between p and x

    Homework Statement (cos(x))^p \leq cos(px) 0\leqx\leqpi half and p, 0\leq(not equal) p \leq(not equal) 1 i need help, if some one can tell me how to started, what should i used i will really apreciate it! (sorry for my english :confused:) Homework Equations The Attempt at...
  22. L

    Sketching the spectrum of a cosine wave

    Homework Statement An amplitude-modulated cosine wave is represented by the formula x(t) = [ 10 + cos(2*pi*(2000)*t) ] * cos(2*pi*10^4*t) Sketch the two-sided spectrum of the signal. Be sure to label important features of the plot. The Attempt at a Solution x(t) = 10 * cos(2*pi*(2000)*t) +...
  23. F

    Superposition of cosine functions for tides

    Homework Statement I have 1.805cos(2pi / 12.165x) +3.125 as the function of tidal data. However I need to use another function, superimpose, to more accurately graph the data for the tides. And yeah I'm pretty lost... Does the next function have to be something to do wit the moon and its...
  24. F

    Cosine function & Modelling Tides

    I'm having a bit of trouble working out the cosine function for the data I have on tide charts. Homework Statement I need to put the data provided into the cosine function y=acos(nx-b)+c Morning average high tide: 5.137 metres Morning average low tide: 1.29 metres Afternoon average high...
  25. A

    Question on linear combinations of sines and cosine (complex analysis)

    I have a question on complex analysis. Given a differential equation, \dfrac{d^2 \psi}{dx^2} + k ^2 \psi = 0 we know that the general solution (before imposing any boundary conditions) is, \psi (x) = A cos(kx)+B sin(kx). Now here's something I don't quite understand. The solution...
  26. N

    Calculate the cosine of OAB

    Relative to a fixed origin O, the point A has position vector 4i + 8j – k, and the point B has position vector 7i + 14j + 5k. (a) Find the vector A to B. (b) Calculate the cosine of OAB. (c) Show that, for all values of t, the point P with position vector ti + 2tj + (2t - 9)k lies on the...
  27. G

    Particle dynamics on a cosine path.

    Hello, I have attached the problem that I am referring to. Basically I have a particle traveling on a path defined by a cosine function and I want the equations of motion with respect to the inertial system, if the particle is under the influence of friction and gravity. I have calculated...
  28. X

    Understanding the Cosine Wave Frequency Domain

    So...a cosine wave when shown in the frequency domain is represented by a 1/2 magnitude vector at the positive frequency and a 1/2 magnitude vector at the negative frequency. I am surprised by this because I cannot understand why a vector of magnitude 1 at the positive frequency of the cosine...
  29. S

    Calculate Area Difference of Triangles ABC with AB=5cm, AC=3.2cm

    soine cosine rule I have a question for you, I came across this question while revising for my exam on monday if anyone can answer this I'll be very impressed. Two different trangles ABC have AB=5cm, AC=3.2cm and angle ABC=35degrees calculate the difference between their areas.
  30. A

    Is There a Straightforward Proof for the Cosine Rule?

    Hi Guys/Girls, I have a maths exam tomorrow and there is a chance I could be asked to prove the cosine rule or sine rule. Well I have a simple proof of the sine rule but cannot find a simple one for the cosine. They all seem very advanced. Would anybody have a straightforward proof for...
  31. A

    How can geometric vectors be used to solve a river crossing problem?

    Homework Statement 1. Homework Statement A river is 2 km wide and flows at 6 km/h. A motor boat that has a speed of 20 km/h in still water heads out from one bank perpendicular to the current. A marina lies directly across the river on the opposite bank. Use Geometric Vectors to solve...
  32. G

    Partial fraction decomposition of the cosine

    Homework Statement Calculate \sum_{n=2}^{\infty}\frac{1}{n^{2}-1} with the "standard" method and with the method of the partial fraction decomposition of the cosine. Homework Equations \pi\cot\pi z=\frac{1}{z}+\sum_{k=1}^{\infty}\frac{2z}{z^{2}-k^{2}} The Attempt at a...
  33. E

    If cosine is equal to -12/13 Find sine and tangent in Quadrant

    If cosine is equal to -12/13... Find sine and tangent in Quadrant If cosine is equal to -12/13... Find sine and tangent in Quadrant Two... What is the answer??
  34. R

    Differentiate the law of Cosine

    It is a simple math question, but I am stuck. The law of cosines is R^2=h^2 + r^2 - 2 h r cos (theta). Theta is of course the angle facing R. To differentiate, I first set r^2 - 2hr cos\Theta+ (h^2-R^2) = 0 2 r \dot{}r - 2 h cos\Theta \dot{}r + (h^2-R^2) = 0. Is there any...
  35. V

    Finding the maximum and minimum points of a cosine function

    Homework Statement This is part of a larger engineering problem, I have reduced it to this mathematical equation, which should be simple, right? I need to find when the following EQ has Maximum's (in terms of x)...
  36. S

    Proving the Relationship between Inverse Sine and Cosine Functions

    To show that cos-1(-x)-cos-1(x)=2sin-1(x) I tried take x= sina taking cos of the whole equation cos(cos-1(-x))-cos(cos-1(x))=2cos(sin-1(x)) now we have to prove : -x-x=2cos(sin-1(x)) LHS: -2x=-2sina=2cos(a+pi/2) RHS: 2cosa Iam not sure how to proceed further..can anyone help me...
  37. R

    Finding sine and cosine formulas

    [SOLVED] Finding sine and cosine formulas Homework Statement If sinx/siny = 1/2 and cosx/cosy = 3 prove: sin (x + y) = 7/3 sinx cosx Homework Equations sin (x + y) = sinx cosy = cosx siny The Attempt at a Solution Can someone please give me a hint so that I can start? Thanks.
  38. W

    Normalization of a wave function with cosine

    I need to normalize the following wave function: psi= Cexp(-abs(x))exp(-iwt)cos(pix) I know that when squaring it, the time dependent part drops out, which is good, but what I seem to be left with is Psi^2=C^2exp(-2abs(x))cos^2(pix) Which seems like a fairly complicated integral to...
  39. P

    Can You Graph Cosine and Sine Functions? A Guide for Beginners

    I need help on graphing cosine and sine functions. i know how to read a graph and come up with the equation but i don't know how to do it the other way around. i want to be able to graph something like y=-2+2cos0.5x
  40. F

    Calculators Not getting the right answers for sine, cosine and tangent with TI-89

    I'm not getting the right answers for sine, cosine and tangent functions. I have no idea why this is.
  41. I

    Sine, Cosine and Tangent Trig Help

    [SOLVED]Sine, Cosine and Tangent Trig Help I'm in 10th grade, I was just doing my homework when this dawned on me: How would one find Sine, Cosine and Tangent without a calculator. So if I am stuck in the desert with a stick I could find the Cosine of 73º by stick and sand method...
  42. T

    Trigonometry, different products of sine and cosine

    Homework Statement There is a right angled triangle, with the following angles, a, b, and 90deg. If a < b, how many different values are there among the following expressions? sin a sin b, sin a cos b, cos a sin b, cos a cos b Homework Equations The Attempt at a...
  43. B

    Understanding when to use sine and cosine to find x and y components

    I am having trouble understanding when to use sine and cosine to find x and y components. I know that its not always going to be the same (ex. you won't always use cosine to find x component.) Any input would be appreciated!
  44. D

    Find angle between vectors with cosine law

    Hi I would really appreciate it if anybody could lead me in the right direction on this one... |A| = |B| |A+B| = 100|A-B| I need to find the angle between A and -B for the statement to be true. Using cosine laws I've come up with the following eqn: |A+B| = 100|A-B| 2|A|^2 + (2|A|^2)(cosx)...
  45. M

    What is a wavefront? sine or cosine graph?

    What is a wavefront? Huygen's principle says that every point on a wavefront acts as a source of wave with the same speed. Now I am not asking you what that means, but i don't understand what a wavefront is. Is that like a circle? But wave is usually like a sine or cosine graph, so what's the...
  46. B

    Prove Hyperbolic Cosine Sum-to-Product Identity

    Homework Statement Prove the identity: Cosh(x) + Cosh(y) = 2Cosh[(x+y)/2]Cosh[(x-y)/2] Homework Equations Cosine sum-to-product http://library.thinkquest.org/17119/media/3_507.gif The Attempt at a Solution Can you use the same formula for Cosine sum to product for hyperbolic...
  47. W

    Geometrical interpretation of Taylor series for sine and cosine?

    I've stumbled upon what might be a geometrical interpretation of Taylor's series for sine and cosine. Instead of deriving the Taylor's series by summing infinite derivatives over factorials, I can derive the same approximation from purely geometrical constructs. I'm wondering if something...
  48. P

    Solving the Expanded Cosine Series of y=sin(x) in (0,180)

    expand the function y=sinx in a series of cosines in the interval (0 to 180) i want to know only the value of f(x) for solving this.what is the value of f(x).
  49. malawi_glenn

    Complex cosine equation (complex analysis)

    Homework Statement Solve cosz = 2i , z\in \mathbb{C} The Attempt at a Solution e^{iz}+e^{-iz} = 4i t=e^{-z} t+t^{-1}=4i \Rightarrow t^{2}-4it+1=0 t = (2 \pm \sqrt{5})i log(e^{-z}) = logt z = x + yi;x,y \in \mathbb{R} log(e^{-z}) = log(e^{-y+ix}) = -y +xi +...
  50. C

    Integral with cosine and exponential - cos (2t) x e^2t

    Homework Statement can som1 please give me an idea where to start with this integral. integral of: cos (2t) x e^2t thanx Homework Equations The Attempt at a Solution
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