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. C

    Related Rates involving Cosine Law

    For problem: See Attachment I've never done a problem of this sort and it's proving to be much more difficult compared to the other problems I have had assigned to me. I'm not entirely sure which formulas to use but I've been playing with the following: Length of Arc = r\theta A = \pir^2 C =...
  2. T

    Orthogonality of sine and cosine question

    Hello, I'm trying to solve Fourier Series, but I have a question. I know that cos(nx) is even and sin(nx) is odd. But what does this mean when I take the integral or sum of cos(nx) or sin(nx)? Do they have a value or do they just keep their form?
  3. N

    Alternative deduction of sum of sine and cosine

    Hi! Many students know that A\sin(x) + B\cos(x) =\sqrt{A^2+B^2} \sin{(x+\arctan \frac{B}{A})}. I have seen just one deduction of that relation, showed by set up a system of two equations, solving for amplitude and phase shift. Is it possible to deduce the relation in a vectorial way, or in...
  4. U

    Cosine perturbation to potential well

    Homework Statement Part (b): Find the perturbed energy. Homework Equations The Attempt at a Solution I've solved everything, except part (b). I got an answer of 0 for part (b) for all orders, which is kind of strange, as one would expect some perturbation. \Delta E_n = \langle \psi_n...
  5. S

    MHB Using Residue Calculus For a General Cosine Angle

    Hi, I am supposed to use residue calculus to do the following integral $$\int_{0}^{2\pi}\frac{1}{a+b\cos( \theta) } \mathrm{d}\theta$$ for |b|<|a| i have paremetrise it on $$\gamma(0;1)$$ that is $$z=\exp(i\theta), 0\leq\theta\leq2\pi$$ and obtain the following...
  6. M

    Why is the term w(t-x/c) used in the cosine representation of a traveling wave?

    This is about wave reflection and transmission. For an infinite string with a density change at x=0, consider an incident wave propagating to the right from x = -∞. The most general form is W = A cos(w(t-x/c)+θ), with amplitude A, angular freuqency w, time t, distance x (from origin), wave...
  7. T

    MHB Due to a symmetry of the cosine we can just double the integral from 0 to 1

    Hello, ∫|cos(px/2)|dx between [0,2] I encountered this rule. How does this apply to other intervals of say [3,4],[7,9] etc. Are the numbers both halved? so [3,4] becomes [1.5,2] etc? Also, does this rule apply to all symmetrical functions? Thank you, Tim
  8. R

    Using a Fourier Cosine Series to evaluate a sum

    Homework Statement a) Show that the Fourier Cosine Series of f(x)=x,\quad 0\leq x<L is x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right] b) use the above series to...
  9. S

    MHB Calculating Fourier Cosine Series of cos(x) from 0 to \pi

    Find the Fourier cosine series of cos(x) from x=0 ~to ~\pi Here the Fourier series is given by f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\inf}a_n cos nx dx where a_n=\frac{2}{\pi}\int_0^\pi f(x)cos nx dx I am facing problem to solve it. I am getting a_0=0 and a_n=0 so the Fourier series becomes...
  10. A

    Limit of cosine question

    Homework Statement What would be the limit of {cos(m!*pi*x}^{2n} as 'n' and 'm' go to infinity along with proof? Also x is real. I have no clue as to how to start the question. Please help! Thanks. The Attempt at a Solution I tried to write the original function as...
  11. T

    Cosine of Vector: Adjacent/Hypothenuse

    Homework Statement The cosine of a vector is equal to the opposite over its magnitude. Homework Equations The attempt at a solution[/b] the cosine is the ratio of the adjacent to the hypothenuse this is why i say it "came' is incorrect.
  12. MarkFL

    MHB Calculate Definite Integral of arcos(tanx) from -pi/4 to pi/4

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  13. B

    Integrating power of a cosine times a complex exponential

    Homework Statement Consider the inner product $$\frac{1}{2\pi}\int_0^{2\pi} \left(\frac{3}{5 - 4\cos(x)}\right) e^{-ikx} dx, \quad k \in \mathbb{Z}, \quad x \in \mathbb{R}.$$ Homework Equations Is there a method to solve this without using the residue theorem, e.g. integration by parts...
  14. A

    Why cosine wave for phase deviation in phase modulation?

    Phase modulation is a system in which the amplitude of the modulated carrier is kept constant, while its phase and rate of phase change are varied by the modulating signal. By the definition of phase modulation, the amount by which the carrier phase is varied from its unmodulated value, called...
  15. P

    Cosine Law: Query About Equation | Online Physics Course

    My online physics course is using cosine law to find the net forces on objects. My question is to do with the equation, at first it shows it as : c^2=a^2+b^2-2ab cosC. From there, it changes to: c=[a^2+b^2-2ab cosC]^1/2. How and why does this work? Why isn't a square root involved in the...
  16. Seydlitz

    N-rowed determinant which corresponds to cosine multiple angle formula

    Homework Statement I need to prove or show that this n-rowed determinant which corresponds to cosine multiple angle formula is in fact true using induction. The Attempt at a Solution First let ##a = \cos \theta## and suppose I have this n by n determinant. $$ \begin{vmatrix} &a &1 &0& \\...
  17. D

    Cosine Similarity: Explained and Examples

    Just for clarification... If I take the cosine similarity of two vectors and i get an answer of 1, then bother vectors are equal and the same. If I do the same again with another two vectors and get an answer of 0, then the vectors are at an angle of 90 degrees to each other and...
  18. N

    What are the smallest four positive solutions for 9cos(2t)=9cos2(t)-4?

    Homework Statement 9cos(2t)=9cos2(t)-4 for all the smallest four positive solutions Homework Equations The Attempt at a Solution I've factored it and pulled out 9cos(t) and made that = u so i have u2-2u-4 That factors to 3.23606 and -1.236 Next i added 9cos(t) back in...
  19. E

    MHB Number of Real Solutions to Cosine System Between 0 and 2π

    How many (unordered) sets of pairwise distinct real numbers \{t_1,t_2,t_3,t_4\} all between 0 and 2\pi are there such that in some order they satisfy the following system: $$\begin{align*}\cos(2t_1)=4\cos(t_1)\cos(t_2)\\ \cos(2t_2)=4\cos(t_2)\cos(t_3)\\ \cos(2t_3)=4\cos(t_3)\cos(t_4)\\...
  20. N

    MATLAB Plotting cosine in matlab looks really funky

    I have to plot a bunch of cosine waves and then add them up but my problem is that each indiviudal wave looks really weird when I plot and not like a cos wave. The program runs and plots the final result but I'm not sure how accurate it is because when I plot each individual cos wave I know it...
  21. E

    Justify an equality involving hyperbolic cosine and Fourier series

    Homework Statement The problem: Justify the following equalities: \cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2} I am trying to figure out how to start this. When I insert the Euler identity of \coth (using...
  22. J

    MHB Fourier Cosine Series: Equivalence for {x}

    2. Fourier cosine series correspondence for f(x)= x, o < x < pi given by x ~ pi / 2 - 4/n, E infinity on top and n=1 on bottom. cos (an-1)/x / (2n-1)squared, (0 < x < pi). Explain why this correspondence is actually an equality for 0 is less than or equal to x and x is less than or equal to...
  23. D

    Discrete Cosine Transform(DCT)

    The N real numbers x0, ..., xN-1 are transformed into the N real numbers X0, ..., XN-1 according to one of the formula by DCT. I would ask what is the benefit and why should we do that?
  24. P

    MHB Completion of the proof of the Cosine Rule

    Hello my friends, I posted this picture as a proof of the Cosine Rule in another thread, however after having a closer look at it, I believe it is incomplete. It works by drawing a segment from one of the vertices so that this segment is perpendicular to one side of the triangle, and then...
  25. K

    MHB Partial Derivatives of the cosine rule.

    Partial Derivatives Hi all I was wondering if anyone could help me with this problem. I have a triangle that has a = 13.5m, b = 24.6m c, and theta = 105.6 degrees. Can someone remind me of what the cosine rule is? Also (my question is here) From the cosine rule i need to find: the...
  26. Y

    Help with sine and cosine integral

    I am verifying the equation of radiation power of dipole antenna. I found mistakes in the derivation in the notes. I know the final equation is correct. So instead of following the steps in the notes, I reverse the step by using the final formula and going back step by step. Here is the final...
  27. MarkFL

    MHB Chloe's question at Yahoo Answers involving the angle sum identity for cosine

    Here is the question: Here is a link to the question: Help with precalculus! Sum or difference formula? - Yahoo! Answers I have posted a link there to this question so the OP can find my response.
  28. Y

    What is the importance of studying Sine and Cosine Integral?

    I want to study a little bit more of Sine and Cosine Integral. I looked through all my textbooks including Calculus, ODE, PDE, Linear Algebra...Nothing! I found info on the web, but mostly are definitions. Where is this subject belongs to? Anyone can give me a link to a more complete...
  29. Telemachus

    Cosine theorem and maclaurin expansion

    Hi. I have a doubt about an exercise in a book of optics. It's about Youngs double slit experiment. The exercise asks to apply the law of cosines. That part was easy, you can see in the diagram, alpha is the complementary angle for theta, it goes straight forward. What I got is this...
  30. Q

    Exploring the Concept of Cosine Tuning

    Hi, I've been listening to the podcast "Neuroscientists Talk Shop" and heard some participants repeatedly refer to something called "cosine tuning" in relation to predicting the efferent reactions to certain clusters of neurons (e.g. a simian moving in a certain way). [sorry if that's...
  31. L

    Simplifying a cosine + cosine with conjugate denominators

    Homework Statement -\frac{1}{2}[cos(\frac{\pi+\pi n}{\pi+\pi n}) + cos(\frac{\pi-\pi n}{\pi-\pi n})] Homework Equations cos(u)cos(v) = \frac{1}{2} cos(u+v)+cos(u-v) The Attempt at a Solution I am attempting to use the above trig function to simplify the first function, but I can't seem to...
  32. P

    Need help determining whether to use cosine or sine

    I'm having trouble understanding when to use sine or cosine in my physics problems for example I got this from one of the lectures off youtube and I know why they are using sine and cos because it is at an angle but I'm rotating the angle to understand why it is mgcosbeta and why the other is...
  33. Petrus

    MHB Help with Solving Limit of Cosine and Sine

    Hello, I got problem solving $$\lim_{x \to 0}\frac{\cos(x)-1}{\sin(x)}$$ I do not really get any progress, I would be glad if someone could give me tips!
  34. R

    Proof the identities of the sine and cosine sum of angles

    Homework Statement I just have to prove the well known identities: \cos(\alpha + \beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin( \beta) But the thing is that I've to use the Taylor power series for the sine and cosine...
  35. F

    Addition of Sine & Cosine

    Homework Statement Express 3sin(ωt) + 2cos(ωt) in the form Rsin(ωt + α) AND verify the resultant function is the same frequency as 3sin(ωt) and 2cos(ωt) Homework Equations R = √a2+b2 α = arctan(b/a) The Attempt at a Solution My attempt using the equations above produces...
  36. F

    Question regarding Cosine Function

    Do anyone know what properties of cosine function should we use? and the method to use it? kinda blur...
  37. A

    What is the connection between sine and cosine and geometry?

    Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x2 And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and...
  38. N

    Has anyone else developed a hate for cosine?

    It's always a brat to differentiate/integrate it due to the sign change. Why can't it be more like sine?
  39. W

    Find the cosine of the angle between the normals to the planes

    1. Homework Statement Find the cosine of the angle between the normals to the planes: x+y+2z=3 and 2x-y+2z=5 2. Homework Equations [/b] x+y+2z=3 and 2x-y+2z=5 3. The Attempt at a Solution All I know is cos θ= V * W / ||V|| ||W||
  40. L

    MHB Proving the Limit of Cosine Squared: $\mathbb{Q}$ vs. Non-$\mathbb{Q}$

    \lim_{n\to\infty} \left (\lim_{k\to\infty} \cos (\left| n! \pi x\right|) ^{2k} \right) = \begin{cases} 1&x \in \mathbb{Q} \\0& x \not\in \mathbb{Q}\end{cases} . How to prove it?
  41. T

    Understanding the Cosine Rule: Deriving the Other Two Formulas

    So here is my question, I understand how to derive the cosine rule from, both triangles acute and obtuse. My problem is the 3 formula you get from this equation. When I derive from a triangle I get the formula: c^2=a^2+b^2-2abcos∅ so how do you derive the other two formula, I read that...
  42. G

    How Do You Integrate Cosine Powers in Calculus?

    Homework Statement \int { { cos }^{ 2n+1 }(x)dx } Homework Equations { cos }^{ 2 }+{ sin }^{ 2 } = 1 The Attempt at a Solution i got to here: \int { { (1-{ sin }^{ 2 }(x)) }^{ n }d(sin(x)) } Any help would be appreciated!
  43. S

    Minimize a certain function involving sine and cosine

    Homework Statement It isn't a homework problem per se, but a curiosity a stumbled upon when trying to solve a physics problem (I was trying to calculate the angle I would need to do less work possible, while moving the box). The equation I found is: f(\theta)=\cos(\theta)+...
  44. Jalo

    What is the Cosine Fourier Transform of an Exponential Function?

    Homework Statement Find the cosine Fourier transform of the function f(t)=e-at Homework Equations The Attempt at a Solution F(w)=(2/π)0.5∫dt e-atcos(wt) The integral is from 0 to +∞ Using euler's formula I got the result F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a)...
  45. S

    What is the Cosine of a 2 by 2 Matrix?

    I have to find the cosine of the following: | 1 0 | | 2 2 | This is a 2 by 2 matrix. I have been reading my linear algebra book for awhile now, and theirs no information at all on how do to something like this. Not sure how to find the cosine of a matrix. Could not really find any pages...
  46. Z

    Simplification of answer involving Cosine

    I have manged to get my answer down to the first line in the picture but I have tried all ways and can't seem to simplify it to the second line. Thank you
  47. C

    Understanding the Cosine Inverse Function and the Cast Rule

    I am currently doing a trig question and trying to find the location of the acute angles of cosine inverse and I am just wondering if cosine inverse follow the cast rule? In other words would it be positive where cos would be positive or not?
  48. M

    What does phase of the motion in terms of cosine displacement mean?

    What does "phase of the motion in terms of cosine displacement" mean? I'm getting tripped up on the wording of this homework question. Homework Statement Measured acceleration: An accelerometer has measured the simple harmonic motion shown in the image below. Homework Equations You...
  49. E

    So, for example, if θ = 2π, then θ = 2π ≠ cos-1(cos(2π)) = cos-1(1) = 0.

    Homework Statement cosθ = 1.316581Homework Equations cos-1cosθ = θ // if and only if 0\leqθ\leq1 The Attempt at a Solution This problem is actually the result of an attempt at a solution to solve an even larger problem (SAS triangle specifically). I need to know how to restrict the cosθ...
  50. W

    How to calculate inverse cosine of two variables

    Hi, all I am looking into inverse cosine operations. I have a question like follows: Let x and y be two variables of degrees, how to separate equation arccos(x+y) into an equation that contains x and y separately? Such as arccos(x+y) = f1(...x) + f2(...y)? Thank you very much for your...
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