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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Can Bessel Functions and Cosine be Expressed as Infinite Series?

    Homework Statement Show that \cos x=J_{0}+2\sum(-1)^{n}J_{2n} where the summation range from n=1 to +inf Homework Equations Taylor series for cosine? series expression for bessel function? The Attempt at a Solution My approach is to start from R.H.S. I would like to express all...
  2. F

    Phase and Magnitude of a Cosine

    Homework Statement Graph the magnitude and phase of the function: H(w) = cos(3w) Homework Equations None The Attempt at a Solution So here's the thing, I understand how to graph the phase and magnitude of any sort of function like X(w) = A*exp(wt). In that case, the magnitude...
  3. A

    Fitting a Cosine Function to Inflection & Known Derivative Point

    I am trying to fit a cosine function to two points knowing that the first is an inflection point (e.g. a trough) and also knowing the gradient at the second. I have a gut feeling this has a unique solution it just needs the right identities and massaging but as of yet I haven't found the way...
  4. P

    Simplfying Inverse Hyperbolic Cosine

    Homework Statement Simplify the following expression: arccosh \left(\frac{1}{\sqrt{1 - x^2}}\right) \forall x ∈ (-1, 1) Homework Equations cosh(u) = \left(\frac{1}{\sqrt{1 - tanh^{2}u}}\right) u ∈ ℝ The Attempt at a Solution x = tanhu ∴ u = arctanhx u ∈...
  5. mnb96

    Relationship between hyperbolic cosine and cosine

    Hello, I am considering the hyperbola x^2-y^2=1 and its intersection with the line y=mx. The positive x-coordinate of the intersection is given by: x=\sqrt{\frac{1}{1-\tan^2\alpha}}=\sqrt{\frac{\cos^2 \alpha}{\cos(2\alpha)}}=\cos\alpha \sqrt{\sec(2\alpha)} where we used the identity...
  6. M

    Hyperbolic cosine looks like a parabola

    Hello, I wanted to know why the graph of the hyperbolic cosine function (1/2(e^x)+1/2(e^-x)) looks like a parabola. Is there any reason for this? I suppose the individual exponential functions both go to infinity in a symmetric way... but I wanted a better reason :). Thanks, Mathguy
  7. M

    Linearising a Cosine Curve: Exploring the Range of g-forces

    Homework Statement For my physics EEI, I have developed the formula: g-forces=√(391.88-337.12 cosθ)/9.8 I need to linearise the graph into the form y=mx+c. I'm not sure where just the angle is the independent or cos of the angle. Homework Equations y=k√(x) can be graphed as y vs...
  8. M

    What is the Fourier Cosine Integral Identity for Deriving B* and A(w)?

    Homework Statement show that xf(x)=integral from 0 to infinity of [B*(w)sin(wx)]dw , // B* is a function not B * w where B* = -dA/dw A(w) = 2/pi integral from 0 to infinity [f(v) cos(wv)] dv Homework Equationsf(x)=integral from 0 to infinity [A(w)cos(wx)] dw The Attempt at a Solution...
  9. D

    MHB Special cases for sine and cosine sum

    State the special cases of the above two formulas for $n = 0, 1,$ and $2$. These should be familiar formulas. I don't see what is so special and familiar about when n = 2 or for cosine n = 1.When $n = 0$, we have $$ \sum\limits_{k = 0}^0\cos k\theta =...
  10. D

    MHB Proving Sum of Cosines Simplifies to Trig Identity

    Prove that the $\sum\limits_{k = 0}^n\cos k\theta = \text{Re}\left(\frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}\right)$ simplifies to $$ \sum\limits_{k = 0}^n\cos k\theta = \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta $$ So I have that the real part...
  11. H

    What is the method for solving cosine squared equations?

    I was just finishing my physics homework (don't worry this isn't a HW question, the homework is done) and the last calculation I had to do was cos2(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right. Now I'm curious though. How would one solve this by hand?
  12. O

    Finding Triangle Angles with Cosine Rule

    For part (i), my answer is correct but my answer for (ii) seems to be a little bit out. I can't spot where I've gone wrong. Can anyone help me out? Many thanks. Homework Statement Q. In the given triangle, find (i) |\angle abc|, (ii) |\angle bac|. The Attempt at a Solution (i) cos B =...
  13. P

    Proof of difference identities for cosine

    Hi, I am working on proofs of the difference identities for sine, cosine, and tangent. I am hoping to solve these using a specific diagram (attached). I was wondering if you could help me with the difference of cosines. Is it possible to derive it using the attached diagram? If so, how...
  14. D

    Sum of Geometric Series with cosine?

    Homework Statement With a series like: pi^(n/2)*cos(n*pi) How am I meant to approach this? Do I use the Squeeze Theorem? Homework Equations The Attempt at a Solution
  15. D

    Why sine is used for cross product and cosine for dot product?

    While we calculate cross product of two vectors let A and B we write ABsinθ. And while we calculate dot product of them we write ABcosθ. Why particularly we use sinθ for cross product and cosθ for dot product.Is there any physical reason why we choose sine for cross product and cosine for...
  16. X

    Vectors Using Cosine or Sin Law

    Homework Statement Need help with a question ... .__." q: if the magnitude of a=2 and the magnitude of 5a-2b= 7.7, and the angles between the vectors a and b is 50 degrees determine the magnitude of b. Homework Equations The Attempt at a Solution first I drew a diagram...
  17. T

    Fourier Cosine Series Coefficient Calculation: A_n = (2)(-1)^{n+1}(n≥1)

    Homework Statement The Attempt at a Solution So after I have properly extended the series, I want to find the A_n coefficients. In the case of A_0 I get 1/2 -- good, great. That's what the book gets. Now for the general A_n: A_n = \frac{1}{2} \int^2_{-2} (1)cos(\frac{n\pi x}{2})...
  18. Z

    PAM signal with a cosine input

    sketch the spectrum of the PAM signal S(f) if the input is m(t)=cos(2*pi*fm*t) where fm=3000 at a sampling rate of 10000 using rectangular pulses of duration 0.04ms. in the range +-15Khz S(f)=H(f)*M'(f) taking the Fourier transform of the rectangular pulse we obtain: H(f)=Tsinc(Tf) where...
  19. D

    Question about the start of a cosine fourier series

    Question about the "start" of a cosine Fourier series Hey. I was just looking through Paul's Online Notes http://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx to teach myself Fourier Series and I had a question about the a_{0} term of the cosine series. In the online lesson...
  20. Ƒ

    Cosine for synchronous demodulation

    Homework Statement Synchronous demodulate x(t). Homework Equations xc(t) = cos(2pi*fc*t), fc is the carrier frequency xm(t) = cos(2pi*fm*t), fm is the modulation frequency x(t) = xc(t)*(1+m*xm(t)), m is the modulation index m = .8 fc = 2000 hz fm = 200 hz The Attempt at a Solution I...
  21. H

    Drawing (not too simple) Cosine waves on the x and y axis.

    I have to draw cosine waves in relation to pattern formation for mathematical biology, for example, I have to plot things similar to these on the x-axis; cos( 2*pi*x / √5 ) from x= 0 to √5 cos( 3*pi*x / 2 √5 ) from x= 0 to 2√5 cos( 3*pi*x / 2 √(5/6) ) from x=0 to 2√5 And with the y-axis; cos(...
  22. L

    Equations with sine and cosine.

    Homework Statement Solve for |θ1-θ2| Homework Equations cosθ1 + cosθ2 = 0 sinθ1 + sinθ2 = 0 The Attempt at a Solution This is a silly math problem within a larger question I'm working on. I have solved for it multiple times now using different trig identities and I get different answers...
  23. H

    Uniform Convergence of Fourier sine and cosine series

    Homework Statement f(x)= {1, ‐1/2<x≤1/2} {0, ‐1<x≤ ‐1/2 or 1/2<x≤1} State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞ Homework Equations The Attempt at a Solution...
  24. H

    Solve Cosine IND Limit Without L'Hospital

    It's the following one: \displaystyle\lim_{x \to{0}}{\frac{1-\cos(1-\cos x)}{3x^4}} In case we have to apply L'Hospital, appart from it, how could I solve this without it? Thanks!
  25. Matt Benesi

    Continuity of real portion of cosine variant of 3d fractals

    By continuity I mean an unbroken fractal. With certain variants, one ends up with sharp gaps in the fractal. mag=({x^2+y^2+z^2})^{n/2} yzmag=\sqrt{y^2+z^2} \theta= n *atan2 \;\;(x + i\;\;yzmag ) \phi = n* atan2\;\; (y + iz) new_x= \cos{(theta)}\;*\;mag new_y=...
  26. A

    What is your method of knowing when to use sine and cosine in force problems?

    I am having the hardest time attaching my brain to some sort of method to know when to use sine and cosine on force problems. What is an easy way of remembering which function to use to find the force in the direction of x and force in the direction of y?
  27. S

    Integrating squares of sine & cosine

    Homework Statement find ∫4cosx*sin^2 x.dx Homework Equations The Attempt at a Solution ∫4cos x * 1/2 (1 - cos2x) ∫2cosx - 4cos^2 x. Then i don't know whereto go from here??
  28. M

    Calculating Limits Involving Cosine: Is My Approach Correct?

    Homework Statement find the limit of : \lim_{x\rightarrow0}\frac{\sqrt{5-cos(x)}-2}{x^{2}} Homework Equations The Attempt at a Solution I multiplied the numerator and the denominator by the conjugate of the numerator and i got : \frac{1-cos(x)}{x^{2}(\sqrt{5-cos(x)}+2)} then: i divided...
  29. fluidistic

    Heat equation, Fourier cosine transform

    Homework Statement Problem 8-17 from Mathew's and Walker's book: Use a cosine transform with respect to y to find the steady-state temperature distribution in a semi-infinite solid x>0 when the temperature on the surface x=0 is unity for -a<y<a and zero outside this strip. Homework...
  30. D

    Fourier sine and cosine tranformation, difficult problem, (for me)

    Homework Statement What are the Fourier sine and cosine transformations of exp(5t)? Homework Equations Fc (ω) = (√(2/∏))∫exp(5t)cos(ωt)dt , (between boundaries of infinity and zero) The Attempt at a Solution When I try to integrate by parts I just end up going round in...
  31. H

    Finding Cosine of an Angle in 3D space

    Homework Statement a) Find the cosine of the angles that the line r = [-3,2,5] + t [2,2,√2 ] makes with the coordinate axes. b) If a,b and c are the angles that the line makes with the x,y and z axis respectively, find the value of cos^2a + cos^2b + cos^2c. c) What is the magnitude of the...
  32. B

    What is the Solution to Simplifying Trig Expressions?

    Homework Statement Homework Equations The Attempt at a Solution I don't see how the textbook gets from step 1 to step 2. If anything, the cosines cancel and the answer should be (sine^2)x
  33. S

    Hyperbolic cosine identity help

    Homework Statement Show that cosh^2(x) = (cosh(2x) - 1)/2 Homework Equations cosh(x) = (e^x + e^-x)/2 The Attempt at a Solution I have attempted this multiple times and get the same results every time. Squaring cosh(x) I get 1/4(e^2x + e^-2x +2), which is i believe 1/4(cosh(2x) +2)...
  34. M

    Finding the nth Derivative of Cosine Function

    Homework Statement \frac{d^{2n}}{dx^{2n}}\cos x n \in N Homework Equations \cos x=\sum^{\infty}_{k=0}(-1)^k\frac{x^{2k}}{(2k)!} The Attempt at a Solution \frac{d^{2n}}{dx^{2n}}x^{2n}=(2n)! But k is different that n. I don't have a clue how to solve that.
  35. S

    How to Determine the Distance Between Deer Using Vectors and Cosine Law?

    Vector and cosine law?? Homework Statement Three deer, A, B, and C, are grazing in a field. deer B is located 62m from deer A at an angle of 51 north of west. deer C is located 77 degree north of east relative to deer A. The distance between deer B and C is 95m. What is the distance between...
  36. C

    Solving Sinusoidal Equations involving Cosine Inverse

    Suppose that a spaceship is fired into orbit from Cae Canerveral. Ten minutes after it leaves Cape, it reaches its farthest distance north of the equator, 4000 kilometers. Half a cycle later it reaches its farthest distance south of the equator (on the other side of the Earth, of course!), also...
  37. J

    Modeling Tidal Changes with Cosine Functions

    Homework Statement High tide at 4am with a depth of 6 meters. Low tide at 10 am with a depth of 2 meters. Model the problem using the equation to show the depth of the water t hours after midnight. Homework Equations y= A cos(Bx+C) +D The Attempt at a Solution: I am not getting...
  38. E

    Simplifying cosine and sine expressions

    Homework Statement So I am a little rusty on my basic math and I am trying to simplify this expression. In case you were curious the expression came from a signal block diagram that I calculated the transfer function H(z) to then found the frequency response H(f) and I think i have done that...
  39. P

    Question involving integration and cosine

    Homework Statement Show that, if n is an odd number, \int_0^\pi \cos^nx dx = 0 Homework Equations The Attempt at a Solution \int_0^\pi \cos^nx dx = \int_0^\pi \cos^{n-1}(x)\cos (x) dx = = \int_0^\pi (\cos^2x)^{\frac{n-1}{2}} \cos x dx = \int_0^\pi (1 - \sin^2x)^{\frac{n-1}{2}} \cos...
  40. C

    Cosine Rule Help: Solving the Cut Size of a Curved Bar

    Homework Statement Hello! I've run into a problem at work and need a quick solution! Basically I need to work out the cut size of a curved bar. I have the chord length (650mm) and the radius' (1335mm) Obviously i need to calculate the inner most angle, then multiply my diameter by pi, divide...
  41. N

    Cosine Rule Problems - Solve Arbitrarily Chosen Triangles

    Hey, If I wanted to make up problems that are solved using the law of cosines, shouldn't it work out even if I arbitrarily choose side a, b, and θ? After all, any two sides and an angle between them form a triangle. Correct?
  42. Z

    Orthogonality of cosine and sine functions

    Can someone give a more intuitive explanation on how it is (if it is true), that; ∫all cos (nx) cos (mx) = 0 if n!=m or ∫all sin (nx) sin (mx) = 0 if n!=m thanks
  43. V

    Different rotation matrix, with cosine?

    I know that a proper orthogonal rotation matrix in R^{2} has the form [cos \theta sin \theta -sin \theta cos \theta] which would rotate a vector by the angle \theta. However, I have also seen the matrix [sin \theta cos \theta -cos \theta sin \theta] What type of rotation...
  44. Z

    Summation involving sine and cosine

    Homework Statement \omega^2=(2/M)\sum_{n>0}\frac{A\sin(nk_0a)}{na}(1-\cos(nKa)) A, a, and k_0 are constants, n is an integer. I need to find \omega^2 and \frac{\partial\omega^2}{\partial K}, but I have no idea where to start.Homework Equations Not sure, the stuff above.The Attempt at a...
  45. N

    Find the Fourier cosine series of f(x)=x(Pi+x)

    Homework Statement Find the Fourier cosine series representation of g(\chi) = \chi (\pi + \chi) on the interval (0,\pi) The attempt at a solution Okay so I've got a0=\frac{1}{\pi}\int\chi(\pi+\chi)d\chi =\frac{5\pi^{3}}{6} an=\frac{1}{\pi}\int\chi(\pi+\chi)cos(n\chi)d\chi for...
  46. G

    A Cosine Law question involving angle of depression

    Smart people help! Trignometric question. Homework Statement A pedestrian bridge is build over a river. The angle of depression from one end of the bridge to a large rock beside the river is 37°. The distance from that end of the bridge (ptA) to the rock is 112m while the distance from the...
  47. B

    Cosine Theta in Work: Force Angle Relation

    Hello, I was wondering if the cosine of theta, in the work equation, related to the angle of the applied force?
  48. I

    Find two solutions for Cosine theta = 1/2 for

    Find two solutions for Cosine theta = 1/2 for... [b]1. Find two solutions for Cosine theta = 1/2 for 0 degrees less than or equal to theta less than or equal to 360 degrees. Express answers in degrees and radians. Homework Equations [b]3. I know that Cosine theta = 1/2 gives you 60...
  49. M

    Net Force on Log: Solving Using Cosine & Sine Laws

    Homework Statement Two ropes are attached to a log that is floating in the water. A force of 80.0 N is applied to one rope and a force of 60.0 N is applied to the other rope, which is lying at an angle of 40° from the first rope. What is the net force on the log? We know that the angle...
  50. M

    Why do sine and cosine have different x intercept patterns?

    hi everyone. Not really a homework question but I'm trying to teach myself trig and I wonder Why does a sine graph have x intercepts in multiples of pi and why does a cosine graph have intercepts pi/2
Back
Top