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. T

    Using the Cosine Rule: When Do You Get 2 Answers?

    Hi, I recall that when you use the cosine rule you can sometimes come out with 2 answers (somthing to do with cos graph?). I can't qutie remeber and I've looked through my books and I can't find it. I can use the rule to calculate one side or angle, but when do you get 2 answers? Can someone...
  2. D

    Mastering Trigonometry: Understanding Sine, Cosine, and Tan Graphs

    I'm having a lot of problems with this topic. I know what the sine, cosine and tan graphs look like. one question i come across fequently is "given that sin 30°, what is a) sin 150° b) sin 330°. - i can work it out on a calculator but the questions on a non-calc paper. I am presuming it's...
  3. L

    Area enclosed by sine and cosine.

    Homework Statement Hello, I'm trying to find the area enclosed by sine the cosine function on the interval 45 degrees and 225 degrees, my problem is i get a negative number after i do the integration, my answer is -2 root 2. here's what i did, sin(x)-cos(x)dx after integrating...
  4. M

    Simple cosine expression NOT so simple

    :grumpy: I am rediscovering my long lost "A" level maths and have been having lots of fun working on puzzles. Unfortunately, I seem to have hit something of a wall in the following simple expression that is repelling every attack I launch: f(x)=Sqrt(1+kcos(x)) I have tried substituting...
  5. J

    Roller Coaster, Cosine 'ramp', Exit Velocity?

    Homework Statement The teacher screwed up and gave us a problem that he could not solve. I am trying to solve it for fun. A roller coaster has a cosine shape. The coaster starts at the top (0 pi) with near 0 velocity Friction is 0, and there is no air resistance. Homework Equations...
  6. L

    How to determine the point of intersection of sine and cosine?

    Homework Statement Im not sure how to start this question: determine the points of intersection between y=sin x and y=cos 2 x for x between 0 and pi. The Attempt at a Solution First thing that comes to mind is the eqaute the two, but i don't know how that helps me?
  7. D

    Malus Law - Cosine squared term?

    this is just a general trig question: We are going over Malus law in physics; the formula is this: S = s*cos^2(theta) My question is about the cosine squared term in the equation. does this simply mean take the cosine of a number and square it? in other words would this be the same thing...
  8. W

    What is the Cosine Rule for Triangles with Operands Greater Than 1?

    Homework Statement As part of a Mechanics problem, I need to find the resultant of two forces. I was able to find F[Resultant]'s magnitude easily enough, but it's direction stumps me. ...because when I rearrange the Cosine rule to find angle A, the operand of Arccos is greater than 1...
  9. J

    Cosine Rule , what good is it ?

    Cosine Rule , what good is it ? Does the Cosine rule hold true for say negative lengths ? as in a vector quantity like displacement ? I came across this problem which had -15km and 10km as the known sides whereas the angle opposite the unkown side is 60 degree ... I tries using c^2 =...
  10. J

    Solving a Vector Problem using the Cosine Rule

    Question : Two ships A and B leave port P at the same time . Ship A travels due north at a steady speed of 15km/h and ship B travels N 60 degree E at a steady speed of 10km/h. what is the distance and direction from A to B after 1 hour ? what is the velocity of B relative to A ? Solution ...
  11. S

    Fourier Cosine series of cos(x)

    Hello peoples, I think this is a trick question... well sort of :P http://img133.imageshack.us/img133/472/picture8ox1.png for part (a) i get that the cosine Fourier Series for f(x) = cos(x) to be: http://img138.imageshack.us/img138/6114/picture9sq2.png i hope that is ok, but its...
  12. P

    Constructing Sine & Cosine Series for f(t)=t

    When a question asks Construct sine and cosine series for the function: f(t)=t, 0<t<pi. Should I assume the period of f(t) is pi? I think it must because the domain is discontinous at 0 and pi.
  13. H

    Why Sine is an odd function and Cosine is an even function?

    Hello, I'm curious if anyone can shed some light on my seemingly opaque brain as to why Sine is an odd function and Cosine is an even function?
  14. B

    Cosine Fourier seires of cosh(t-1)

    Hi, can someone give me some help with the following? The cosine Fourier series of period 2 for the function f(t) that takes the form f(t) = cosh(t-1) in the range 0 \le t \le 1 is \cosh \left( {t - 1} \right) = \sinh \left( 1 \right)\left[ {1 + 2\sum\limits_{n = 1}^\infty {\frac{{\cos...
  15. Y

    How can a DCT help identify the obvious frequency of a discrete wave?

    Hello, Suppose I have a discrete function of a perfect cosine wave. So if I will do a DCT on this function I will get one non zero coefficient which corresponds to the perfect cosine wave, and the rest will be zero. Now I have a pass filter, which filters out anything with a frequency which...
  16. R

    How Do You Model Building Sway with Trigonometric Functions?

    this is my question: A Building sways 55cm to the right from origin in 5 seconds and 55 cm to the left of the origin in 35 seconds. And i am supposed to write an eqaution to define this. I'm guessing the is no amplitude no vertical translation and since it's sine basically I am going to...
  17. R

    Understanding Cosine Theta in Work Equation

    Hello Physics Specialists, I am having some difficulty understanding the Work Equation with respect to the Cosine Theta portion of the equation. Work = Force * Distance * Cosine(theta) Attached is a picture of two hypothetical mechanical devices. I understand how to calculate work...
  18. denian

    What do you mean by direction cosine

    what do you mean by "direction cosine" i've just came through this term, and may someone please help clarify what does it mean? tqvm.
  19. dextercioby

    Exploring the Limit of Cosine as x Goes to Infinity

    of doing this \lim_{x\rightarrow +\infty }\cos^{x}\frac{\pi}{x} than using \cos x \simeq 1- \frac{x^{2}}{2} \mbox {when "x" goes to zero} ? Daniel.
  20. B

    Why cosine theta for calculations?

    When figuring the force between 2 points, no angle the result is calculated just by distance (and charge). But when there are 3 points, either 2 charges or more and a point or 3 charges, why is there an angle calculation?
  21. E

    How do you solve for the intersection of a line and a cosine function?

    Suppose you are given the equation of a line, and a given cosine function that the line intersects. How do you solve algebraically, that is non-graphically, for the point of intersection of the line and the cosine function? Inquisitively, Edwin
  22. M

    Understanding the Cosine Law: A Geometric Approach

    hi, Is anybody who can explain the proof of cosine law here? thanks,:smile:
  23. G

    Linear Combination of Cosine Function

    How would I express cos(wt+1) as a linear combination of cos(wt) and sin(wt)?
  24. T

    Discovering the Cosine Series of Sine: Insights and Calculations Explained

    I tried to find the cosine series of the function f(x) = \sin x, using the equation below: S(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) where: a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx I found: a_0 = \frac{4}{\pi} a_n = \frac{2 }{\pi (1 - n^2)} (\cos(n \pi) + 1)...
  25. E

    Does exponential take over cosine?

    If i have a response for a circuit that consists of: cos(t) + e-t + e3tand if I let t->infinity, does exponential "over-take" cos(t)? well, the actual question is whether the output/response remains finite as t->infinity... thanks, as always :smile:
  26. T

    Sine, cosine and tangency of angle 11pi/12

    When I worked this problem, I came up with the following: sin 11pi/12= -sq root 2/4 (sq root3-1) cos 11pi/12= -sq root 2/4 (sq root3+1) tan 11pi/12= 2-sq root3 am I far off?
  27. K

    Inverse cosine with varriables

    Is there any equation or method that can be used in place of a trigonomic function when side values of a triangle are known only as varriables? For example: triangle abc where c is the center of a circle and AC and BC are radiui who's value = x, and AB = x-2y. So far as I know, the area of...
  28. P

    Understanding Sine, Cosine and Tangent Angles

    Hi people, what is meant by sine, cosine, tangent angles, apart from their general definition of opp side/ hypo side; adj side/ hyp side and so on? how are these sine, cosines and tangent angles invented by humans?
  29. S

    Proving Inequality with Cosine Rule and Schwartz Proof

    Hi, i was required to show that -1 < \frac{a.b}{\|{a}\|\|{b}\|}} > -1 I did this by using the cosine rule which is c^2 = a^2 + b^2 - 2a.b\cos{\vartheta} How ever our teacher did it by a scharts proof which i don't quite understand, :mad: , Now my question is why can't i prove it...
  30. M

    Solving an Integral Involving Conversion to Cosine

    i had this problem on an exam today, Intagral of (x^2)/(4+x^2)^(7/2) i arrived at a point where i had to convert the integral of 1/secx to cos, is that correct?
  31. A

    Sine and cosine law in oblique triangles

    Word Problem The leaning tower of pisa leans toward the south at an angle of 5.5 degrees. One day a shadow was 90 m long and the elevation from the tip of the shadow to the top of the tower was 32 degrees 1)Determine the slant height of the tower. First I found all the angles and then...
  32. M

    Approximating the Cosine Integral?

    Does anyone know of a semi-quick way of approximating Ci(x)? I tried to find an asymptotic expansion for it, but had little luck. Truth be told, I'm not even sure exactly what the definition of asymptotic expansion is. I discovered it while learning about ways of approximating harmonic numbers...
  33. A

    Solving for Cotangent using Cosine and Pi

    \cos \theta = \frac {2} {3} and \frac {3\pi} {2} <\theta<2\pi then determine the exact value of \frac {1} {\cot\theta} I did this question using sin^2+cos^2=1 subbing in cos and then solving for sin when i got both values i realized that 1/cot theta = tan theta which is (sin theta)/(cos...
  34. A

    Solving a Triangle with the Cosine Law: Help!

    I just read about the cosine law and the sine law. I have a practise problem and know to use the cosine law but what ever answer I get gives me a math error in my calculator. the sides are a=4.3 b=5.2 c=7.5 I need to solve the triangle so find the 3 angles within. a^2=b^2+c^2-2bc...
  35. U

    Solving a Tower Height with Cosine Law

    Question: A man is looking at the top of a tower. The angle of elevation to the top of the tower is 10 degrees. 100m closer to the tower, a man has an angle of elevation to the top of the tower of 20 degrees, how tall is the tower? My Problem: I can solve this ver easily by recognizing...
  36. M

    How to find the cosine between the directions

    I really need some help on this one: You push a box up a ramp using a horizontal 100-N force F. For each 5m of distance along the ramp the box gains 3m of height. Find the work done by F for each 5m it move along the ramp (a) by directly computing the dot product from the components of f and...
  37. S

    Simplifying with Cosine and Sine

    Simplify in terms of cosine and sine only. tan^2x - {csc^2x\over cot^2x} From here, I assume you can flip the fraction and make it {tan^2x\over sin^2x} next, i reduce it to: (sec^2x-1) - ({sec^2x -1\over 1 - cos^2x}) and anyway..im lost; i don't know where to stop and how...
  38. M

    Visualizing Sine and Cosine on a Unit Circle: Accurate or Not?

    Is this an accurate way of displaying sine and cosine on a unit circle? :wink:
  39. K

    Calculating Sine, Cosine, Tangent & Roots

    Hi...I know what Sine, Cosine, and Tangent represent in math and how they are used...but I'm asking a different question. That question is: how are they calculated? That is, when I put, say Sine(10) into a calculator, how does it calculate that value? Additionally, how are square roots...
  40. Z

    Finding Imaginary Numbers with Cosine of 3

    Question: Use the formula cos(x + iy) = cos(x)cos(iy) - sin(x)sin(iy) to find two imaginary numbers whose cosine is 3 My workings so far: cos(x)cos(iy) - sin(x)sin(iy) = 3 Therefore: cos(x)cosh(y) - i(sin(x)sinh(y)) = 3 cos(x)cosh(y) = 3 sin(x)sinh(y) = 0 sin(x) = 0 x = 0...
  41. L

    What are some real world applications of sine and cosine functions?

    I have a math project due where the assignment is to find real world applications of the sine and cosine funtions. Can anyone recommend some good websites on the web or have any specific ideas? Thanks, -Charlie
  42. C

    Is My Solution for 4sinx - 3cosx Correct?

    I'm unsure as to whether I'm correct with this: 4sinx - 3cosx My answer ended up being 5sin(x + sin^-1 (-3/5)). The equations I used were: asinx + bcosx = (a^2 + b^2)^1/2 * sin(x + angle) angle = sin^-1 (b / (a^2 + b^2)^1/2)) Can someone clarify whether I'm correct? I would...
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