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. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

    Homework Statement Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any...
  2. C

    Finding Resultant Force Using Cosine and Sine Law

    Homework Statement "Calculate the net force with the resultant angle acting on each object indicated in the diagram." The line going off to the southeast is supposed to be straight. My computer art skills aren't great. | 22 N | |_ _ _ _ _ \ ) 35 degrees \ \ 38 N 2...
  3. evinda

    MHB Find cosine series: Period 4 for f(x)

    Hello! I want to find the Fourier series for the given function $f$: $f(x)=\left\{\begin{matrix} 1, & 0<x<1,\\ 0, & 1<x<2 \end{matrix}\right.$ -> cosine series, period 4 I also want to find the graph of the function to which the series converges , for three periods and then make some...
  4. Guidestone

    Why is this rigidity constant decomposed using a squared cosine?

    Good night guys. I've been studying for a test on mechanical vibrations and I came across this problem of a telescopic crane (or whatever its name is). It's necessary to obtain a equivalent constant kp and then it has to be decomposed into its vertical component and the author uses a squared...
  5. J

    MHB Calculate Cosine of a Matrix: Solutions to Systems

    https://uploads.tapatalk-cdn.com/20170308/78feec183e9672f563c5e41b4c52e1d9.jpg https://uploads.tapatalk-cdn.com/20170308/4ad8560adf9e090969c38515a31d1407.jpg Please help, I know the definition of a cosine of a matrix is cos(a) = I-1/2!A^2+1/4!A^4-... But I am unsure how this would help me find...
  6. maxhersch

    I Entries in a direction cosine matrix as derivatives

    This is a somewhat vague question that stems from the entries in a directional cosine matrix and I believe the answer will either be much simpler or much more complicated than I expect. So consider the transformation of an arbitrary vector, v, in ℝ2 from one frame f = {x1 , x2} to a primed...
  7. F

    Solving a Physics Problem: Sine vs. Cosine Theta

    Homework Statement Question is provided The Attempt at a Solution The required solution is provided My question is, the examiner have used sine theta in their expression for change in potential. I keep ending up with 1 - cosine theta Second question is how did they get rid of theta in the...
  8. Drakkith

    Coefficient for a Term in a Taylor Expansion for Cosine

    Homework Statement The coefficient of the term (x−π)2 in the Taylor expansion for f(x)=cos(x) about x=π is: Homework Equations ##cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}...## The Attempt at a Solution Unless my taylor series for cosine is incorrect, I'm...
  9. A

    I Which reduced vector should be used in cosine similarity?

    In Latent semantic analysis, the truncated singular value decomposition (SVD) of a term-document matrix ##A_{mn}## is $$A=U_rS_rV^T_r$$ In many references including wikipedia, the new reduced document column vector in r-space is scaled by the singular value ##S## before comparing it with other...
  10. Liam C

    B What do Sine, Cosine and Tangent do?

    When I press these buttons on my calculator to find the third side or an angle in a triangle, what calculation is happening? What is the logic behind it all? I know it's a very basic question, but I am only in grade 10 and have not started Math yet this year.
  11. Monoxdifly

    MHB Solve [ASK] Limit of Cosine: How to?

    How to solve \lim_{x\rightarrow\frac{\pi}{2}}\frac{\cos{x}}{x-\frac{\pi}{2}}? At first I tried to convert cos x to \frac{\tan{x}}{\sin{x}} but then realized that \lim_{x\rightarrow c}\frac{\tan{x}}{x} only applies if c = 0. So, how?
  12. A

    Probability of cosine of angle between two directions in collision

    The question refers to the Feynman lectures on physics Vol I chapter 39. He discusses collisions between gas molecules. Here is a relevant extract: They are equally likely to go in all directions, but how do we say that? There is of course no likelihood that they will go in any specific...
  13. V

    Components of a force. When do I use cos or sin?

    Here is an example of a problem I am having trouble with. I need to find the i, j, k of A. I have no issues with finding the components for B, but A I just can't wrap my head around when to use cos or sin. Especially here with double projection. I know that A is : (-10 cos70 sin30, 10 cos70...
  14. D

    Sum of Sine and Cosine: Expressing Any Sum as C sin(α+ϕ)

    Homework Statement Show that any sum: Asin(α) + Bcos(α) can be written as : C sin(α+ϕ) 2. Homework Equations The Attempt at a Solution i can express cos(a) as as sin(90-a), and then try to use the formula that adds sines, but it gives the form of cos*sin. [/B]
  15. M

    I The "real" angle between two triangular surfaces

    Hello everyone, i'm new to the forum so hope it is the right place for my question :) i need to know the angle between two triangular surfaces, the easiest way would be extract the normal for each surface(u,v) and then using the dot product we can easily compute the cosine for the angle I'm...
  16. L

    A Saddle point Integrals with logarithm and cosine

    I'm trying to use the saddle point method to solve the following integral: Z = (1/sqrt{2 pi t}) ∫_{1}^{infinity} ds (1/sqrt{2 pi s}) exp{ p [-s ln(s/t) +s] } cos(2 pi L~ p ~ s), as p → infinity Mod edit to make integral more readable: $$Z = \frac{1}{\sqrt{2\pi t}} \int_1^{\infty} \frac 1...
  17. J

    A Convergence of a cosine sequence in Banach space

    Does the sequence \{f_n\}=\{\cos{(2nt)}\} converge or diverge in Banach space C(-1,1) endowed with the sup-norm ||f||_{\infty} = \text{sup}_{t\in (-1,1)}|f(t)| ? At first glance my intuition is that this sequence should diverge because cosine is a period function. But how to really prove...
  18. M

    Complex Fourier Series into a Cosine Series

    Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...
  19. RJLiberator

    Help solving fourier cosine series related problem

    Homework Statement I am doing #9. Homework EquationsThe Attempt at a Solution I've been looking at a lot of similar problems on the internet. The main difference between this one and them is that this one has an interval of [0,4] while they often have intervals of [0,pi] or [-pi,pi] In my...
  20. UMath1

    Can Trig Substitution with Cosine be Used Instead of Sine?

    I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint...
  21. RJLiberator

    Quantum Measurements: The Average value is the cosine

    Homework Statement We know: In a measurement of quantum mechanics (basic) the average value is the cosine of the angle between the orthonormal basis of the measurement apparatus and the qubit (vector) entering it. Question: How do we prove this?Homework EquationsThe Attempt at a Solution Can...
  22. C

    Cosine and sine of 2Pi/5 problem

    We have a complex number ω = cos(2π/5) +isin(2π/5) and we have two complex numbers a and b, such that: a= ω + ω4 and b= ω2 + ω3. I have to prove that a + b = -1 and a*b= -1. Then, based on that, determine cos(2π/5) and sin(2π/5). I've tried to solve this using trigonometry. First a+b, i got...
  23. Ravendark

    Integral with sine, cosine, and rational function

    Homework Statement I would like to compute the following integral: I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta} where ##a,b \in \mathbb{R}_+##. 2. The attempt at a solution Substitution ##x = \cos \theta## yields I = \int\limits_{-1}^1...
  24. S

    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...
  25. O

    Continuous Time Fourier Series of cosine equation

    Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...
  26. A

    Linearising Cosine: Taking Roots of Equation

    Homework Statement Linearise the following equation I(theta) = (I1 – I2 ) cos (2theta) + I2 where I1 and I2 are constants. Homework Equations I(theta) = (I1 – I2 ) cos (2theta) + I2 The Attempt at a Solution Not sure how to linearise the cos(2theta). Have tried trig identities, would the best...
  27. L

    Work & Energy: Forces with Angles

    Homework Statement A student could either push or pull, at an angle of 30 degrees from the horizontal, a 40kg crate, where the coefficient of kinetic friction is .21. The crate is moved 18m. Calculate the minimum work for pushing and pulling. Homework Equations W=F•(change in)X•cos(angle in...
  28. M

    What is the Inverse Cosine of a Squared Angle?

    Not sure if I'm doing this right. I have an angle theta to find but the cosine has been squared. I brought over inverse cosine to multiply to leave theta on its own. I was told the answer should be closer to 37 degrees? Am I doing something wrong here?
  29. M

    Factoring Cosine: Get Help Solving the Problem

    The Question: My Attempt: I use the 'Double Angle Formula' to get to the second line. Then multiply the sigma-x into what's in the brackets, then factor out at the end. I'm not sure how I reach the answer that's provided with the solution. Any help would be appreciated. Thank You.
  30. J

    B What does cosx/x and tanx/x represent?

    What do the functions cosx/x and tanx/x represent?
  31. maverick_76

    Exploring the Limit of Cosine at Infinity in Integrals

    okay so I have this integral: 2∫cos(kx)dk The bounds are from zero to infinity, this doesn't converge but is there any other way to describe this?
  32. Stephanus

    Understanding the Limits and Derivatives of Sine and Cosine Functions

    Dear PF Forum, In previous threads, I have asked about sine and cosine. The answer given by the members/mentors/advisor are very clear. But lengthy. Perhaps these yes/no questions that I can simply remember and not forget it (again). So here we are 1. if h = 0 then sin(h) = 0 2. if ##\lim_{h...
  33. Greg

    MHB Trigonometry challenge - cosine product

    Prove \cos20^\circ\cdot\cos40^\circ\cdot\cos80^\circ=\frac18
  34. Shahab Mirza

    How to solve this Cosine Law Equation?

    Question is regarding Scalars and Vectors Article. Q: One of the two forces is double the other and their resultant is equal to the greater force . The angle between them is ? Ans : Its answer is cos^-1 (-1/4) My solution : The formula for Cosine law is R= √A²+B² +2ABcos theta Our teacher...
  35. M

    Why do we have both sine and cosine?

    This might seem like a really basic question that one might cover in gr 9 or 10 but instead my friend and I were discussing it now, when he just got his degree and I'm a credit away from mine: why on Earth are there both sine and cosine functions when simply one would do? Either can be...
  36. CAH

    Cosine Rule Problem: I Can't Do a(ii) - Get Help Here

    See the photo attachments of question and marking scheme and my attempt at a solution :) I've done a(i) but I can't do a(ii). Thanks
  37. G

    Prove arccos(-x)+arccos(x) = pi

    Homework Statement From an old edition of the Anton calculus text, I am asked to prove cos-1(-x) + cos-1(x) = π or equivalently cos-1(-x) = π - cos-1(x) Homework EquationsThe Attempt at a Solution Earlier I proved that sin-1(x) was odd by noting sin-1(sin(x)) = -sin-1(sin(-x)), so I...
  38. A

    Finding the orthonormal basis for cosine function

    Homework Statement si(t) = √(((2*E)/T)*cos(2*π*fc*t + i*(π/4))) for 0≤t≤T and 0 otherwise. Where i = 1, 2, 3, 4 and fc = nc/T, for some fixed integer nc. What is the dimensionality, N, of the space spanned by this set of signal? Find a set of orthonormal basis functions to represent this set of...
  39. Q

    Direction cosine matrix of rolling disk on circular ring

    Hey all, I'm stuck on this problem and not sure how to proceed/if I'm in the right direction. Problem: One reference frame N sits at the origin (inertial frame) while another frame, B, describes a disk rolling on a circular ring about the other frame. Picture below (A) find the direction...
  40. D

    Superposition of two cosine waves

    Homework Statement Superposition of two cosine waves with different periods and different amplitudes. Homework Equations This is basically: acos(y*t) + bcos(x*t) The Attempt at a Solution I looked at different trig functions but it seems it is not a standard solution. I've found solutions...
  41. P

    Assigning cosine and sines for axes

    Homework Statement Hey everyone, I'm horrible at physics so bear with me if it's a really easy question. I was working a problem out, where I assigned my x-axis with cosine, and my y-axis with sine, like how I always thought it should be. However, in the solution, they assigned sine for the...
  42. J

    Fourier COSINE Transform (solving PDE - Laplace Equation)

    I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...
  43. J

    What is the output of two cosine functions?

    Homework Statement inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t). Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t] Homework Equations included in upload of attempted solution. Trig identities. The Attempt at a Solution Uploaded in pdf. A lot more has been done on the...
  44. L

    Vectors: why is the cosine of this angle always -1/2?

    Homework Statement Pick any numbers that add to x + y + z = 0. Find the angle between your vector v = (x, y, z) and the vector w = ( z, x, y). Challenge question: explain why v.w/|v||w| is always -1/2. Homework EquationsThe Attempt at a Solution I chose (1, -2, 1) for the first part, which is...
  45. P

    Find the fourier sine series of cosine.

    Homework Statement Hi, so I am doing some past exam papers and there was this question; Homework EquationsThe Attempt at a Solution a0 and an both are equal to zero, this leaves only bn. Since you can only use the sine series for an odd function, and cos(t) is even, does this mean i have to...
  46. B

    Gravitational acceleration, cosine problem

    Homework Statement The gravitational acceleration at latitude x (0<x<90) can be estimated with g(x)=a*cos(2x)+b. 1) Determine what a and b is if the gravitational acceleration is 9.780m/s^2 at x=0 and 9.832m/s^2 at x=90. Homework EquationsThe Attempt at a Solution So I begin by entering what...
  47. L

    Cosine question. Scalar product.

    Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...
  48. E

    How Do You Solve for the Adjacent Side Using Cosine?

    [Note: Image orientation fixed, image trimmed to just relevant section - gneill] Homework Statement hypotenuse is 10 opposite is 2 adjacent is unknown ( obviously needed for cosine) Homework Equations Cosine= adjacent/hypotenuse Sine=opposite/hypotenuse The Attempt at a Solution
  49. G

    Finding Min Value of Sum of Cosine Angles in 3D Space

    Dear Friends! How can I find the minimum value of sum of cosine of angles between three unit vectors in three dimension space
  50. R

    Dimensional Analysis: Inverse Cosine

    Homework Statement For the following dimensional equation, find the base dimensions of the parameter f: M M-3 = a cos( f L ) Homework Equations M represents mass, a represents acceleration due to gravity, in terms of mass * length over seconds squared [[M * L]/[t2]] where L represents length...
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