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. G

    Find limits of sine and cosine functions

    Homework Statement Find the limit: lim (1-cos2x)/(xsinx) x->0 Homework Equations Identities The Attempt at a Solution I've done this over and over and over again! The answer is supposed to be 0 but I keep getting 2 ): lim (1-cos2x)/(xsinx) x->0 lim...
  2. P

    Cosine of 2 times theta

    Homework Statement From angles pi minus theta1 to pi plus theta1, what is the integral of cosine of 2 theta times d theta? Homework Equations The Attempt at a Solution When I evaluate this definite integral from pi minus theta1 to pi plus theta1, I get positive one half sine of...
  3. M

    Discover the Cosine and Tangent Demonstration for cos(x)² = 1/1+tan(x)²

    Demontration of: cos(x)² = 1 \overline{1+tan(x)²} Anyone know? If you don't understand: cos(x)² = 1/1+tan(x)²
  4. S

    Taylor Series for Cosine and Accuracy of Calculating Cosine 2

    Homework Statement How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000 The Attempt at a Solution I have said that |Rn(2)| = |cosn+1(a) 2n+1/(n+1)!|<2n+1/(n+1!) Now i can't do it ...
  5. J

    Modeling a Cosine Wave with Six Flags Data: Equations and Solutions

    Homework Statement I have a problem, it's last major grade for my pre cal, it's about six flags and here's the information: H(min)(Trough): 3.ft H(max)(crest): 32 ft b= 10 ft period = 160 ft write a cosine equation to model of the wave Assuming still that the leftmost crest intersects...
  6. D

    Is there a spectrum difference in amplitude modulation with a sine or cosine?

    If I modulate a pulse X(t) with a.) a sine wave or b.) a cosine wave, I have the frequency spectrum expressions a.) \frac{1}{2j}[X(f-f_0)-X(f+f_0)] b.) \frac{1}{2}[X(f-f_0)+X(f+f_0)] When I plot these for a pulse, I see a difference in the magnitude spectrum, but I should not expect...
  7. P

    Isotropic average of a cosine function

    Hi, please look at the following equation. \frac{3}{16}\frac{\nu_{Q}^{2}}{(1+K_{iso})\nu_{0}} \left(\frac{7}{2} \cos^{4}\theta - 3\cos^{2}\theta + \frac{5}{6}\right) In the paper I am reading, this is simplified considering the isotropic average of a cosine function to...
  8. M

    Solving for Cosine & Sine Sum/Difference of Two Angles: Confusing?

    I have yet to see or understand the benefit of knowing the cosine and sine sum and difference of two angles identity... can someone please explain why you would want to break up one angle into two angles... then solve from there... does not make sense to me... just use a calculator.
  9. icystrike

    Fourier Cosine Series for f(t)=1-t; 0≤t≤π | Homework Solution and Equations

    Homework Statement f(t)=1-t ; 0≤t≤\pi Homework Equations The Attempt at a Solution a_{0}=\frac{1}{2}\int_{0}^{2}\left( 1-t\right) dt=0 a_{n}=\int_{0}^{2}\left(1-t\right ) cos(\frac{n\pi\left t \right}{2}) dt = \frac{4}{(n\pi)^2}(1-(-1)^n) Ans that is given to me is: \frac{2}{\pi} +...
  10. L

    Cosine, Sin, Tangent when find force/tension

    Ok... so I had this down and now I am all confused ;/ I am not posting in the homework sections because its not for homework although I will gie an example of a problem... I just want to understand why/how they use these to find the force/tension... Example: A 20 kg loudspeaker is...
  11. S

    How do i distinguish cosine and sine functions

    How do i distinguish between a cosine and sine function simply by looking at the graph? Usually its easy because the graph for the base sine and cosine functions have certain distinctive features(like cosine function intercepts the y-axis at 1 usually) and the sine function hits the origin(0,0)...
  12. H

    Apostol's Theorem 2.5, sine cosine example

    Homework Statement In Example 3 under Theorem 2.5 of Apostol's Calculus Volume 1, I don't understand how the final formula on the right is obtained.Homework Equations The identity cos 2x = 1 - 2 sin2 x implies sin2x = 1/2(1 - cos 2x) so, from Example 2, we obtain: \int^a_0\,sin^2\,x\,dx =...
  13. S

    Sine and cosine functions Zeros

    How Do i find the zeros for this particular sine function? y=asin(k(x-d))+c when we are given a K value i know how to find the zero but when we are not, i don't know how to find the zero(Atleast after everything has been horizontally, and vertically shifted and streched) y=2sin(x-30...
  14. S

    Check my solution please (Sine and cosine functions)

    I was asked to draw the graph of y=cosx-2 This equation means the graph of y=cosx is shifted 2 units down correct? (Since -2 is the C value because if it were the d value in y=acos(k(x-d))+c it would be in degrees, so we shift each y coordinate down 2 units, or subtract the y value by 2, i...
  15. S

    How can you use a table of values to graph y=cosx-2?

    Graph the function y=cosx-2I have already graphed this, there are several methods to solve this, i opted to use table of values to find all the coordinates to y=cosx, then moved all the y coordinates of each point 2 units down, and the y int was at -1...would this method result in the correct...
  16. A

    What is Hyperbolic Cosine Used for?

    I just learned about hyperbolic functions in my calculus class, and though my professor attempted to explain the use of hyperbolic functions, he really did not go very far into it, just providing a weak example ("If two people are holding a chain, cosh(x) factors into how much the chain...
  17. C

    How to Find the Linearization of the Cosine Function at a Given Point

    Homework Statement Find the linearization L(x) of f(x) = cos(x) at a = π/2 Homework Equations L(x) = f(a) + f'(a)(x-a) The Attempt at a Solution I just want to make sure I did this correctly: L(x) = cos(π/2) + -sin(π/2)(x-(π/2)) L(x) = 0-1(x-(π/2)) L(x) = -x + (π/2)...
  18. C

    Find the period of Cosine of Quadratic function

    Hi all, Hope some here can help me with this math problem. Given, y1 = ax^2 + b. y2 = cos (y1). where a and b are constants. Is y2 periodic with respect to x.? Visually using example grpah, seems to be periodic. How do u find the exact period of such a function? Thanks in advance...
  19. jegues

    Cosine Series Sketch and Calculate

    Homework Statement See figure attached for problem statement as well as my attempt. Homework Equations The Attempt at a Solution I'm not fully convinced I've done the sketch of the cosine series correctly or not, are there any mistakes? Also, when solving for, a_{n} I changed...
  20. S

    Using the Cosine Law on Right and Oblique Triangles

    Can the cosine law be used on right triangles as well as oblique triangles?
  21. M

    Unpacking Trig Ratios: Understanding Sine, Cosine, Tan, and More

    Homework Statement What is the importance of knowing how to express trig ratios in terms of sine, cosine, tan, cosec, sec, or cot? Homework Equations The Attempt at a Solution
  22. D

    Is the Integral of Cosine Function Always Zero?

    Hi. I have been experimenting a little to come up with the following "conjecture" \int_0^{2\pi}d\phi f(a+b\cos\phi)\sin\phi=0 where a and b are arbitrary constants and f(x) is any function. Is this true? I guess it can be shown by expanding f in a power series of cosines?
  23. V

    Half range fourier cosine series

    Homework Statement The function f(x) is defined on the interval 0<x<L by f(x)=x. It can be represented by the Fourier cosine series f(x) = a_0 + sum a_n cos(n*pi*x / L) Find its Fourier coefficients a_0 and a_n. Homework Equations Multiply both sides by cos(n*pi*x / L) and...
  24. D

    Lambert's Cosine Law, Computing Diffuse Reflection

    Lambert's Cosine Law, Computing Diffuse Reflection : I have 2 coordinates (5, 6, 0) the light source and (2.3, 1.92, 0) the plane. The diffuse coefficient is 0.6 and the light source intensity is 200. So using the Lambert's Cosine Law, I need to take 0.6 * 200 * the dot product of the...
  25. silvermane

    Finding formulas for sine and cosine functions:

    Homework Statement Find simple formulas for 1+ cos(θ) + cos(2θ) + cos(3θ) + ... + cos(nθ) and sin(θ) + sin(2θ) + sin(3θ) + ... + sin(nθ) The Attempt at a Solution It's not really a homework question, but more for making a problem that I'm trying to solve a little bit more simple...
  26. M

    Clarification regarding the use of Lambert's Cosine Law

    Homework Statement A light bulb is used to light a bunker 10 feet below. A chair sits on the floor of the bunker 3 feet from a spot directly below the bulb. What is the illumination on the floor around the chair if the luminous intensity is 150 candles?Homework Equations * Lambert's Cosine...
  27. J

    Limit of a cosine function

    Homework Statement limit as t-->10 of cos (1/10-t) Homework Equations The Attempt at a Solution Obviously we can't substitute in t = 10 or it would be undefined, so how do we do this. Is there an inequalitly that I haven't considered?
  28. Q

    Period of a sum of cosine signals

    Homework Statement Find the period of the signal x(t) = 2 + 4cos(40Pi*t) + 3cos(60Pi*t) + 4cos(120Pi*t). Homework Equations The Attempt at a Solution The fundamental frequency (fo) = 10 Hz, since that's the greatest common factor of all the frequencies of the cosine signals. So...
  29. D

    Understanding Wave Reflection and Phase Shifts

    I've attached the multiple choice question. The first time round that I did it, I simply just did a phase shift of the whole graph by pi. Basically I just translated the whole graph by pi, resulting in option (D). However, it later came to my attention that by continuing to draw the...
  30. P

    Solving Trigonometric Problems: Exploring Sine, Cosine & Tangent

    hi every one, i have one doubt i studied abt trignomentry. there finding the triangle angle or side of the triangle using sine function. if we are taking right angle triangle sine A = opp/hypo, cos A = adj/hypo and tan A=opp/adj. here we are finding angle for A only why we are having three...
  31. B

    Calculate angle from sine and cosine

    sin x = 0.5299 cos x = 0.8480 Without using inverse cos or inverse sin, is it possible to calculate the angle?
  32. H

    Representing sum of cosine and sine as a single cosine expression

    a.cos(wt) + b.sin(wt) = M.cos(wt + ϕ) Can you give me M and ϕ in terms of a and b?
  33. B

    Equation with inverse cosine needed to be rewritten

    Homework Statement I need to rewrite this equation as a function of y. would love some help since i failed on the right side of the equation. v= [πr(3R-y)y²]/3R + L√[(2R-y)y(y-R)+R²cos^-1(1- y/R)] i mean is it even possible to get y out of there, with inverse cosine and all? Thanks in...
  34. N

    Cosine = Contraction? (Banach)

    So in Analysis I we explained the convergence of cos to a fixed value by Banach's contraction theorem. But is the cos a strict contraction? Is that obvious? (What is its contraction factor?)
  35. D

    Hyperbolic Cosine curve fitting

    Homework Statement I need to fit a curve using cosh to a hyperbola with a vertex of (0,0) and a point at (4,7). The scanned worksheet can be found here http://img519.imageshack.us/i/scan0001gu.jpg/" http://img192.imageshack.us/i/scan0002uz.jpg/" Homework Equations y=a cosh...
  36. H

    Integral of f times cosine, as period of cosine goes to zero.

    Homework Statement The problem is as follows: Let f be a real valued function that is Riemann integrable on [a,b]. Show that \lim_{\lambda \rightarrow \infty} \int_{a}^{b} f(x)\cos(\lambda x)dx = 0 . Homework Equations I am freely able to use the fact that the product of...
  37. G

    Where Did I Go Wrong with the Cosine Formula?

    I got this question wrong and I'm hoping someone can explain to me where I went wrong. In a triangle... angle A = 350 side b = 7 cm side c = 3 cm I need to find side a which is the side opposite angle A. The formaula to use is -a2 = b2 + c2 - 2bc cos A so, with the figures plugged...
  38. Q

    Why is there a sin term in the particular solution for cosine equation?

    Homework Statement Q'' + 100Q' +50000Q = 4000cos(100t) i found the general solution to be e-50t[Acos(50sqrt19)t +Bsin(50sqrt19)t] but i have a problem with the particular solution i tried Cei100t did i try the wrong expression? because when i compared coefficients, i found...
  39. B

    Sum of two cosine functions with angular frequences

    Homework Statement ok so I am given that I(t) = A cos (w1 t)cos(w2 t) where w2<<w1 then I am asked to express I as the sum of two cosine functions with angular frequences P and Q which I have: I = A/2 (cosPt + cosQt) where P = w1+w2 and Q=w1-w2 Im then asked to evaluate the...
  40. T

    Extending f(x) as an Even Function: Obtain Cosine Fourier Series

    Homework Statement f(x) = sin(x) for 0\leqx<\pi. Extend f(x) as an even function . Obtain a cosine Fourier series for f. Homework Equations a_{0}/2 + \sum a_{n}cos(nx) The Attempt at a Solution So as far as I know, to extend sin(x) as an even function you have to make f(x)=-sin(x)...
  41. L

    Using the Fourier Cosine Series for Integral Calculation

    Homework Statement Using the Fourier cosine series for \[f(x) = \left\{ \begin{array}{l} 1,x = 0 \\ 10,x = \pi \\ x,x \in (0,2\pi ) - \left\{ {0,\pi } \right\} \\ \end{array} \right.\], find a series that converges to \[\int\limits_0^{2\pi } {{x^2}dx} \] The Attempt at a...
  42. M

    Can I Add Sine and Cosine Functions with a Non-Factorable Scalar?

    Homework Statement Hi guys, I don't know if this should go here because it is an excerpt from a higher level problem. The part where I get stuck is when I try to add the cosine functions. Is there any way to add sine and cosine functions that have a scalar in front that cannot be factored...
  43. grooveactiva

    Interference of 3 Cosine Waves: Can They Cancel Out?

    Homework Statement 3 waves are represented by these 3 waves inteferring: cos(Θ-π), cos(Θ+π), 3cos(Θ)? cos(\theta-\pi), cos(\theta+\pi), 3cos(\theta). If I want to diagram these, does the cos(\theta-\pi) and cos(\theta+\pi), cancel each other out? Do I need to convert them to sine...
  44. A

    Trigonometric Identities for Sine and Cosine

    Homework Statement 4 \ sin \ \theta \ = \ 3 \ csc\ \theta The Attempt at a Solution sin\ \theta \ = \ \frac {3}{4} \ csc \ \theta sin^2 \ \theta \ = \ \frac {3}{4} sin \ \theta \ = \ \pm \ \frac {\sqrt{3}}{2} 30 \ \deg \ in \ QI, \ 150 \ \deg \ in \ QII, \ 210 \ \deg \ in \...
  45. P

    Solving the Cosine Identity: cos(α-β)cos(α+β) = cos2α - sin2 β

    Homework Statement cos(α − β)cos(α + β) = cos2α - sin2 β Homework Equations cos(α + β) = cos α cos β − sin α sin β cos(α − β) = cos α cos β + sin α sin β The Attempt at a Solution I worked out the LHS which makes it cos2α cos2β - sin2α sin2β=RHS Then, I'm stuck, however, i...
  46. B

    Cosine and Sine rules to get magnitude and direction of a resultant force

    Use the cosine and sine rules to determine the magnitude and direction of the resultant of a force of 11 kN acting at an angle of 50 degrees to the horizontal and a force of 6 kN acting at an angle of -30 degrees to the horizontal. helppp please
  47. S

    Finding Vector Length & Cosine of Angle for Fixed \theta

    Homework Statement Three units vectors a, b, and c have property that the angle between any two is a fixed angle \theta (i) find in terms of \theta the length of the vector v = a + b + c (ii) find the largest possible value of \theta (iii) find the cosine of the angle \beta between a and...
  48. S

    Finding a fourier series for a cosine function

    Hi I have been working on a example and have worked it out as this Homework Statement f(x) = \( \left( \left \begin{array}{ccc}0 & \mathrm{for} & -\pi < x \leq 0 \\ cos(x) & \mathrm{for} & 0 < x < \pi\end{array} where f is defined on the interval ]-\pi,\pi[. Find the...
  49. D

    Explain why cosine formula is always -0,5.

    Homework Statement Pick any numbers that add to: x + y + z = 0 Find the angle between your vector \textbf{v} = (x, y, z) and the vecor \textbf{w} = (z, x, y) Explain why \textbf{v}\bullet\textbf{w} / ||\textbf{v}||||\textbf{w}|| is always -\frac{1}{2} Homework Equations Cosine...
  50. S

    Question about Direction Cosine Matrix

    Hi Everyone, I am facing a basic level problem about direction cosine matrix as I am not an expert in mathmatics. Hope you people will help me.. I am working on inertial navigation system in three coordinate frames : 1. ENU 2. ECEF (XYZ) 3. LLH (Latitude , Longitude , Height) I...
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