Trig functions Definition and 218 Threads

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. Saracen Rue

    I Why is the integral of ##\arcsin(\sin(x))## so divisive?

    Initially, I was attempting to find the function which expresses the area under enclosed between the function ##\arcsin(\sin(x))## and the ##x##-axis (so technically I am looking for ##\int_{0}^{x} \arcsin(\sin(t)) dt## specifically, but got caught up on finding the general antiderivative)...
  2. T

    A Trig functions and the gyroscope

    Good Morning As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel. I stumble on terms like: 1. sin(theta)cos(theta) - cos(theta)sin(theta) 2. Cos_squared +...
  3. N

    Find an equivalent equation involving trig functions

    Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0## ##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0## ##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here. Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0## ##\sin(x)\cos(x) +...
  4. Mayhem

    I Adding trig functions with different amplitudes

    The trig identities for adding trig functions can be seen: But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change? Analysis: If the amplitudes do show up on the RHS, we would expect them...
  5. pairofstrings

    B Cosine of 1 degree and cosine of 60 degrees?

    Why is cos (1)° = 0.9998? cos(60)° = ½? Thanks.
  6. kshitij

    Limit calculation involving log and trig functions

    This was the question, The above solution is the one that I got originally by the question setters, Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand), As you can see that both these methods give different answers...
  7. J

    Can the limit of a quotient of trig functions approach a specific value?

    Hello. Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist. But can a quotient of the two acutally approach a certain value? lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...
  8. xyz_1965

    MHB How do trigonometric functions and their inverses relate to each other?

    Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)? Why does arcsin (sin x) = x? Can it be that trig functions and their inverse undo each other?
  9. R

    Calculating the half maximum point of a function

    This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
  10. R

    MHB Trig Functions: When Plugging in x Returns x

    I have the statement \sin[\sin^{-1}(x)] = x \hspace{7pt} if -1 \leq x \leq 1. How can I tell if plugging in x will return x for \cos[\cos^{-1}(x)] and \tan[\tan^{-1}(x)] ? What if the positions of the regular and inverse functions were reversed? For example, \cos^{-1}[\cos(x)]. I am only...
  11. C

    What Exactly is Conceptual Physics?

    I have been wondering, what is conceptual physics? I remember taking a class in high school that was physics oriented, for example two trains leave a station at different speeds, and arrive at a central point, where do they overlap. Also there were trig functions on how to find the height of a...
  12. opus

    Inverse Trig Functions and Reciprocals

    Homework Statement Evaluate and express your answer in radians: $$cot^{-1}\left(1\right)$$ Homework EquationsThe Attempt at a Solution I start by identifying that the domain of Arccotangent is all real numbers. So 1 is in the domain. From here, I looked at the unit circle and saw that...
  13. Jeviah

    Solving Simpson's Rule: Finding cos^2(1^2)

    Homework Statement hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
  14. Amlan mihir

    Find the value of the trigonometric sum

    Homework Statement i have no idea how to proceed
  15. alijan kk

    Understanding the Derivative of Inverse Trig Functions

    Homework Statement why this formula works ? Homework EquationsThe Attempt at a Solution when i take the derivative of the right side ,,, there is an additional "a" in the numerator in place of 1,, why the derivative of arcsine of (u/a) not exactly same with the expression under the integral sign
  16. OldWorldBlues

    B How Can Trigonometry Calculate Distances in Surveying?

    Hi there! I haven't yet taken a trigonometry course (I'm in High-school), but I have an amateur interest in surveying. Recently I began thinking about how I could calculate the height of a point relative to me, or the distance of the object from me. Naturally, I immediately thought of the...
  17. M

    Proper usage of trig functions in force problems

    Not a particular problem to wonder about but more of a general question, when one has a free body diagram, when is it best to use sine and when is it best to use cosine? I am reviewing some of my tests for a final, and having previously re-read my forces chapter, I thought that angles rising...
  18. DeathbyGreen

    I Infinite series of trigonometric terms

    I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...
  19. B

    Solutions to Equations Involving Exponential and Trig Functions

    Homework Statement Show that ##e^x = x## does not have any solutions, and show that ##\sec x = e^{-x^2}## has only one solution. Homework EquationsThe Attempt at a Solution Here is my proof of the first proposition: Since ##e^x## is concave up on ##\Bbb{R}##, it must lie above all of its...
  20. G

    B Add two functions, same frequency to produce one greater?

    Is their any way to add two wave functions like sin or cos in such a way that you could double the frequency or at least increase it?
  21. F

    Need help solving this trig equation

    Homework Statement what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114 Homework Equations (sinθ)^2 + (cosθ)^2 = 1 The Attempt at a Solution I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
  22. U

    I How to solve this system of equations of trig functions

    I've written it out and it seems impossible. I get -50(sin^2(alpha)) = 86.63 cos(alpha) sin(alpha) - 6.54. Where would I go from there?
  23. SSGD

    I Solve 2*C*sin(W)+P*cos(N*W)=P for W

    Is there a way to solve for W in the below equation. There has to be multiple solution for W, but I am at a loss as to how to solve this. 2*C*sin(W)+P*cos(N*W)=P or 2*C/P*sin(W)+cos(N*W)=1 C and P are constants N is an integer
  24. Erenjaeger

    Which of these relations are functions of x on R

    Mentor note: moved to homework section y = sin(x) y = cos(x) y = tan(x) y = csc(x) y = sec(x) y = cot(x) (a) 0 (b) 4 (c) 6 (d) 2 I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2
  25. Ryan Hardt

    Calculating Uncertainty for a Chain of Trig Functions

    Homework Statement I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below. I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the...
  26. U

    MHB Find the derivative using implicit differentiation (with inverse trig functions)

    Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...
  27. I

    High School Level Physics Homework

    Homework Statement A highway is to be built between two towns, one of which lies 35.0km south and 72.0km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed with respect to due west. Homework...
  28. A

    Finding the Domain of a Trigonometric Function

    Homework Statement Find the domain of this function and check with your graphing calculator: f(x)=(1+cosx)/(1-cos2x) Homework EquationsThe Attempt at a Solution i get to (1+cosx)/(1+cosx)(1-cosx) which is factored. so then setting each one to zero one at a time i figure out that cosx = -1 and...
  29. Alanay

    How do I calculate inverse trig functions?

    On the paper I'm reading the arctan of 35 over 65 is approx. 28.30degrees. When I use the Google calculator "arctan(35/65)" gives me 0.493941369 rad. What am I doing wrong?
  30. C

    Easiest way to learn exact values for trig functions?

    I'm realizing now how much I need to know the exact values of various trigonometric functions, as shown in various trig tables. Memorizing is pretty arduous, and I'd prefer to understand it, so how can I learn all of these?
  31. B

    Exponentials or trig functions for finite square well?

    How do you know when to use exponentials and trig functions when solving for the wave function in a finite square well? I know you can do both, but is there some way to tell before hand which method will make the problem easier? Does it have something to do with parity?
  32. S

    Integral of trig functions over a period

    Can somebody please explain to me why the integral of, for instance, cos((2*pi*x)/a)*cos((4*pi*x)/a) vanishes over the interval 0 to a? As I understand it, this is generally the case when integrating sines and cosines with different arguments "over the interval of a period." But I'm confused...
  33. Amrator

    Rate of Change Using Inverse Trig Functions

    Homework Statement A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...
  34. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  35. D

    Arguments of exponential and trig functions

    What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
  36. V

    Find Intersections of Trig Functions with different periods

    There are 2 trig functions on the same set of axis. f(x)=600sin(2π3(x−0.25))+1000 and f(x)=600sin(2π7(x))+500 How do I go about finding the points of intersections of the two graphs? This was from a test I had recently and didn't do too well on,so any help would be much appreciated. I started...
  37. Chrono G. Xay

    'Wheel-like' Mathematics (Modulating Trig Functions?)

    As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...
  38. T

    Trig functions in terms of x,y, and r?

    I work a good deal better when the equation is in x and y form, is it possible to set up a trig expression like 5Cos(x)/(Sin(x)-1)and substitute the proper x or y equivalent so long as I remember to replace the trig identities later when the problem is finished? Or can you just not solve these...
  39. J

    Rolls Theorem (trig functions)

    Homework Statement f(x) = sin5x ; [π/5,2π/5] finding the point c which f'(x) =0. I understand the theorem and how to complete it, my issue is using the triq functions Homework Equations f'(x) = 5cos5x The Attempt at a Solution 5cos5x=0 cos5x=0 5x=π/3 x=π/15 my answer is not correct, I am...
  40. T

    Evaluate the integral (inverse trig functions)

    Homework Statement [23/4, 2] 4/(x√(x4-4)) Homework Equations ∫ du/(u√(u2 - a2)) = 1/a(sec-1(u/a) + c The Attempt at a Solution I first multiplied the whole thing by x/x. This made the problem: 4x/(x2√(x4 - 4)) Then I did a u substitution making u = x2. Therefore, du = 2xdx. I multiplied by...
  41. S

    Complex number problem with trig functions

    Homework Statement Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B] Homework Equations 1. z=a+bi 2. re^itheta The Attempt at a Solution I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...
  42. karush

    MHB Integrating a Product of Trig Functions

    $$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$ the ans the TI gave me was $\frac{\sqrt{6}}{4}$ the derivative can by found by the product rule. but really expands the problem so not sure how the $\frac{d}{dx}$ played in this.
  43. K

    MHB Finding Formula without using any trig functions

    Find a formula for g(x)= sin(arccos(4x-1)) without using any trigonometric functions. I have the answer key right in front of me, but i still get how to start it off or the steps in solving these kind of questions or how to do it at all :/ Thanks!
  44. snoopies622

    Why are they called hyperbolic trig functions?

    I know if we set x = \cosh \theta , y = \sinh \theta and graph for all \theta 's, we get a hyperbolic curve since then x^2 - y^2 = 1. But — unlike the case of making a circle by setting x = \cos \theta , y = \sin \theta and graphing all the \theta 's — in the hyperbolic graph the angle...
  45. K

    Finding points of intersection algebraically between 2 trig functions

    So I have several problems that ask me to find all points of intersection algebraically, but I haven't been able to make much headway on most of them. The first problem Homework Statement Find all the points of intersection algebraically of the graphs of ... on the interval [0, 4π]...
  46. F

    Basic Trig Functions: Clarifying Definitions

    I just wanted to clear a couple of things up in terms of strict mathematical definition... Is the correct definition of the trigonometric ratios: cos\varphi=\frac{|x|}{r}, sin\varphi=\frac{|y|}{r} as opposed to: cos\varphi=\frac{x}{r}, sin\varphi=\frac{y}{r} (note the lack of...
  47. T

    Derivative of inverse trig functions

    Homework Statement ln(sec^-1(3x^2 +1)) Homework Equations The Attempt at a Solution 1/sec-1(3x2+1) * 1/(3x2+1)(sqrt(3x2+1)2-1) * 6x Is this correct ?, do I just simplify from here ?
  48. N

    Interchanging between real part of cmplx exp and trig functions

    Homework Statement Given u(x,t) = sum( e^(-at/2)*cos(n*pi*x/2L) * Re[A_n*e^(i*w_n*t)+B_n*e^(-i*w_n*t)], and the boundary conditions u(-L)=u(L)=0 for all t; du/dt = 0 for all x at t = 0; u(x,t=0) = e^(-|x|/l) Find A_n and B_n. Homework Equations N/A The Attempt at a Solution I have...
  49. T

    How to find the at rest position of a particle when trig functions

    so my given: s(t)=cos(pie8*t/4) took the derivative= velocity function then, v(t)= -pie/4 *sin(pie*t/4) When is the particle at rest? v(t)=0 now, 0= -pie/4 *sin(pie*t/4) im lost here. I know it's very simple I am just over thinking. What do I do from here? thanks
  50. QuantumCurt

    Integration of inverse trig functions

    This is for Calculus II. I've found most of the integrations on inverse trig functions to be pretty simple, but for some reason this one is throwing me off. Homework Statement \int\frac{x+5}{\sqrt{9-(x-3)^2}}dx The Attempt at a Solution I started by breaking the integral up...
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