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. R

    How Do You Integrate tan^2(x) * sec^3(x) dx?

    Homework Statement [PLAIN]http://integrals.wolfram.com/images/integral.gif(tan[x]^2*sec[x]^3)dx[/URL] Homework Equations Whatever you need to complete problemThe Attempt at a Solution This problem has given me so much grief. I have page loads of work but can't figure...
  2. I

    What is the Limit of (1-cos(x))/x as x approaches 0?

    Homework Statement lim (x*sin(x)) / (2-2*cos(x) x-> 0 The Attempt at a Solution I remember the trick to these is to get sinx/x and (1-cosx)/x by multiplying by x/x... When I multiply the expression by x/x I end up with (x^2 sinx)/(2x(1-cosx)) The sinx/x goes to 1...
  3. F

    Summation formula for trig functions

    Does anyone know if there is a summation formula to find the sum of an expression with n as an argument in a trig function? I'm asking this because I'm learning about Fourier series/analysis but it seems that once we have the Fourier series we only sum for n=1,n=2,n=3... We never sum there...
  4. A

    Insanely easy question - trig functions, need a quick question answered

    Homework Statement I need to solve the equation cos 5x + cos 3x = cos x Do the first two terms add together to be cos 8x? Then use double angle formula to solve?
  5. T

    Complex numbers with trig functions

    Homework Statement prove: arctanh(z) = 1/2 ln( (1+z)/(1-z) ) Homework Equations cosh z = (ez + e-z)/2 sinh z = (ez - e-z)/2 ez = ex + iy = ex(cosy + siniy) The Attempt at a Solution cosh z / sin hz = ez+e-z/ez-e-z
  6. N

    How to Solve Trigonometric Functions with Arcsec and Pi/4

    Homework Statement cos(ArcSec(-\sqrt2+\frac{\pi}{4}) The Attempt at a Solution cos(ArcSec(-\sqrt2+\frac{\sqrt2}{2}) In order to solve this problem how do l deal with the \frac{\pi}{4}. Is it correct to substitute \frac{\pi}{4} with \frac{\sqrt2}{2}
  7. A

    Solving Trig Functions for Simplifying Equations and Finding Solutions

    Homework Statement I was asked how to go about solving for t Homework Equations 140=10t + sin(t) The Attempt at a Solution I used a graphical solution, is there an easier way
  8. H

    Unraveling the Right Angle Proof of Derivatives of Inverse Trig Functions

    I am still a little early in my math education but we were talking about this the other day. Mine is the right angle proof of the derivatives of inverse trig functions. The first time I went threw them I was just given the rule that the derivative of arcsin is such and such, with no real...
  9. M

    Trig Functions Squared: Solving for cos2 0 = 1/2 with Homework Equations

    Homework Statement cos2 0 = 1/2 (0 for angle) Homework Equations The Attempt at a Solution I've never seen cos, sin, or tan to the power of anything before what do i do with the power?
  10. 3

    How Do You Input Trig Functions into a Graphing Calculator?

    Homework Statement How do I input trig functions in the calculator? An example would be cos(2t) = 0.5. Homework Equations The Attempt at a Solution I don't understand how to start it. How do I input this in? I am not sure how to input it into a graphing calculator. The equation...
  11. S

    Antiderivative of product of trig functions

    Homework Statement Find the antiderivative of sin^4(x)tan^2(x) Homework Equations Trig identities I may have overlooked. The Attempt at a Solution I tried writing the integrand in terms of sin and cos but that didn't seem to lead anywhere. I tried integration by parts since the...
  12. N

    Are Trig Functions Odd or Even? - Exploring Properties and Solving Examples

    Homework Statement I don't know how other people learned trig, but in my book, the first 2 chapters have nothing to do with angles, but are mostly circles and trig functions of real numbers. Does this make sense to anyone? I am in the next chapter already and I know there are other, easier...
  13. L

    Converting Inverse Trig Functions to Degrees

    Homework Statement Hello, I have been doing some trigonometry lately, just trying to get the trig I need for calculus, and the book I am reading goes into inverse trig functions. The book dives into trig equations without giving a very good description of how to find the trig functions. I...
  14. T

    Euler's Formula of Inverse Trig Functions

    Is there a clean exponential formula for inverse trig functions (arcsine, arccosine) like there is for sine and cosine?
  15. M

    Calculus: trig functions anti-derivatives

    Homework Statement Let g'(x)=-17x16* sin(Ax9) - 9Ax25*cos(Ax9) g(1)=143/9 Where A is a real number such that tanA=1/sqrt(80), 0<A<pi/2 Find g(0) The Attempt at a Solution I was able to get the antiderivative of g'(x), so that: g(x) = -x17*sin(Ax9) I got A=6.3794 degrees. BUT...
  16. D

    Implicit Differentiation w/ trig functions check

    Homework Statement Determine dy/dx when y*sin(x2)=5 The Attempt at a Solution y*2xcos(x2) dx/dx + sin(x2)*1 dy/dx = 0 2xy cos(x2)*dy/dx = -sin(x2) dy/dx = -sin(x2) / 2xy cos(x2) dy/dx = -2xy tan(x2)
  17. L

    Derivatives of trig functions and isosceles triangles.

    The base of an isosceles triangle is 20 cm and the altitude is increasing at the rate of 1 cm/min. At what rate is the base angle increasing when the area is 100 cm2? I wasnt really sure where to start on this question so i tried my best at an answer. I'm sure I've gone wrong with this...
  18. J

    Integrating inverse trig functions

    I was working on this problem \int{\frac{x^3}{x^{2}+1}}dx I at first tried to use one of the inverse trig functions but couldn't get the form to match...should I try to use log properties...making the denominaor u and trying to get the numerator 1?
  19. G

    Not sure if i've got the trig functions correct

    I'm doing A2 Edexcel maths and i keep on forgetting the trig functions so can someone take a look and tell me if I've got it right. So far: Sin^2(x)=cos^2(x)=1 so: sin^2(x)=1-cos^2(x) cos^2(x)=1-sin^2(x) this is where i get abit stuck (sin^2(x)+cos^2(x)=1)/cos^2(x) =...
  20. R

    Integrating x√(1+x²) Without Trig Functions

    Hi I am trying to integrate x \sqrt{1+x^2}dx by parts...but it seems to involve trigonometric functions - is it possible to solve this integral without using trig functions? Thx
  21. C

    Can I Simplify This Trig Integral Using u Substitution?

    I'm having a wee bit of a problem with this \int(tan x)^3*(sec x)^3dx could i use a u substitution by pulling out a sec x tan x and using it as a du
  22. S

    Powers or Trig Functions Question - Integral

    Homework Statement Integral of : (sec x)^5 (tan x)^3 dx Homework Equations I am having trouble substituting correctly for this equation and i can't get it to work. I believe it has to do with trig powers ? If anyone could help be step by step. or even just a hint on how to solve it...
  23. D

    Why Does My Calculator Show Different Results for Inverse Sin in Trigonometry?

    [SOLVED] Concavity of Trig Functions I'm working out a problem, but I'm stuck at a spot. The original equation is y = 2 cos x + sin 2x, and yes I did the second derivative, which came out to be cos x (4 sin x + 1) when i equate it to zero, i get cos x = 0 and 4 sin x + 1 = 0...
  24. S

    Finding the Period of a Trig Function without Graphing

    Period of trig functions! This is a rather easy-silly quiestion but i just don't know how to show it. I know how to find the period of trig functions of the form f(x)=Asin(wx+c) etc. i mean i know how to show that the period of this is \frac{2\pi}{w}. However, how would one find the...
  25. I

    Partial Derivatives with Inverse Trig Functions

    [SOLVED] Partial Derivatives with Inverse Trig Functions Homework Statement Show that u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations... \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} given that u = ln(x^{2} + y^{2}) and that v = 2tan^{-1} (y/x) Homework...
  26. J

    Problem involving trig functions

    [SOLVED] problem involving trig functions Homework Statement Your room has a window whose height is 1.5 meters. The bottom edge of the window is 10 cm above your eye level. (See figure) How far away from the window should you stand to get the best view? (“Best view” means the largest visual...
  27. B

    Derivatives of inverse trig functions

    Homework Statement find an equation of the tangent line to graph of f at the indicated point. f(x) = arcsin2x (u'/sqrt 1-u^2)(2) 2. The attempt at a solution (1/sqrt 1-2x^2)(2) I got the answer from calcchat but I don't understand where the 1 and 2 came from?
  28. C

    Limits of fractions of polynomials and trig functions

    I have two... Homework Statement The the limit Homework Equations \lim_{x \rightarrow 1} \frac{1-cosx}{x^2} The Attempt at a Solution I figured to just plug in 1, but I wanted to make sure... Homework Statement Find the limit Homework Equations \lim_{x \rightarrow 3}...
  29. L

    Finding the limit of trig functions

    I'm having a hard time find the limits of these trig functions. Please help me with it. Thanks in advance. 1. \lim_{x \rightarrow \pi}(\frac{\sqrt{x}}{csc x}) From this function I know that csc x= \frac{1}{sin x} which cannot equal 0. X, therefore, cannot equal \pi n where n is any...
  30. D

    Trig functions on complex numbers?

    Out of curiosity, what happens when you try to perform a trig function on a complex number? So, say, sin(4i+3)? Is this undefined since angles are only capable of being real numbers, or is there an agreed behavior for complex numbers? DaveE
  31. O

    I finding the limits of trig functions

    Homework Statement lim cos(beta sign)-1/sin(beta) Beta-0 2. lim sin^2 3t/t^2 t-0 for the first one i tried to use the quotient formula to get the derivative, but still I am not sure i did it correctly and for the second problem, i have no idea what to do Homework Equations...
  32. N

    Series involving trig functions

    Homework Statement http://img87.imageshack.us/img87/6784/dsc07215xt5.jpg http://img72.imageshack.us/img72/9435/dsc07216aa0.jpg Homework Equations None. The Attempt at a Solution Answers for part I are circled, here's parts II-IV. http://img485.imageshack.us/img485/8642/dsc07217uq2.jpg...
  33. C

    Finding Areas of Regions Bounded by Trig Functions Using Integrals

    Homework Statement Find the exact total of the areas bounded by the following functions: f(x) = sinx g(x) = cosx x = 0 x = 2pi Homework Equations the integral of (top equation - bottom equation) The Attempt at a Solution Change the window on the graphing calculator to...
  34. M

    Trig Functions Simplified: How to Solve cosx / (1+sinx) | Expert Guide

    Homework Statement Simplify: cosx / (1+sinx) 2. The attempt at a solution 1 / (sec x)(1+sin x) 1 / (sec x + (sin x / cos x)) 1 / (sec x + tan x) I know that the answer is sec x - tan x but don't know how to get there. Any help would be appreciated.
  35. J

    Adding trig functions for positive output

    Hello All, Here's a strange question for you. Is there any possible way to add two or three sine wave functions together so that the resulting wave never goes below the positive Y axis of the graph? These functions can be whatever frequency or phase necessary, just no vertical offset. I am...
  36. A

    Integrating Trig functions (Namely tan^{2}x)

    I know that \int \tan^{2}x dx= \int \sec^{2}x-1\ dx = \tan x - x + C but i don't completely understand how this is derived. Because of this lack of comprehension, I have no idea what to do with \int \sec^{4}5x\ dx. I went from there to get: \int \sec^{4}5x\ dx= \int [\sec^{2}5x]^{2}\ dx= \int...
  37. M

    Integration and inverse trig functions

    Homework Statement getting confused with integration of trig functions. I am finding the integral of sinhx/1+coshx and I'm not sure how to start. should i use an identity? help is appreciated! Homework Equations possibly an identity of some sort? The Attempt at a Solution ?
  38. P

    Finding force with trig functions

    A shopper pushes a 7.5 kg shopping cart up a 13 degree incline. Find horizontal force needed to give the cart an acceleration of 1.41 m/s^2. I know the answer is 28N, but I haven't yet figured it out. I did F=ma F=(7.5)(1.41)cos13= 10.30 what didn't I do? thanks
  39. F

    What is the Relationship Between Trig Functions and Vector Constants?

    i'm thinking that trig functions are like vector constants when used as expressions. Like, y=cos^2x+3, or y=cosx/sinx I'm meaning to talk about what's meant by a trig function is just a number and how trig is used beyong right triangle problems.
  40. K

    Trig Functions- Could anyone tell me why my working is wrong?

    Trig Functions- Could anyone tell me why my working is wrong?? the question seems simple..although i don't know what i am doing because it is obviously wrong!. thnkas heaps if anyone can help.. solve 2cosx-(sqrt(3))cosx sinx = 0 for (-pie<=x<=pie) what i was doing is dividing through by...
  41. A

    This is a problem that requires solving a trigonometric equation.

    Trig Functions... I don't really understand how my book wants me to approach this problem. And I know that you appreciate work...because this is for my benefit after all...but how exactly would this be worked? Direction: Solve the equation for (theta)... 0 is less than or equal to (theta)...
  42. A

    Trig functions translations and combinations of transfomations word problems

    Hi, I really need help with this question 1) the water depth in a harber is 21m at hight tide, and 11m at low tide. One cycle is completed approximatly every 12h. a) find an equation for the water depth as a function of the time, t hours, after low tide b) Draw a graph 48h after low...
  43. J

    How can I prove the continuity of sin(x) using common properties?

    I’m having trouble showing that Sin(x) is a continuous function. I’m try to show it’s continuous by showing: 0<|x - x_0| < d => |sin(x) - sin(x_0)|<\epsilon Here is what I have done |sin(x)| - |sin(x_0)|<|sin(x) - sin(x_0)|<\epsilon and |sin(x)|<|x| so -|x| < -|sin(x)| => |sin(x)|-...
  44. Hootenanny

    Differentiation of Trig Functions

    The question is as follows: Find \frac{dy}{dx} \; \; \; xtanx \; \; \; dx The standard differential is given in the formula book as f(x) = \tan kx \Rightarrow f'(x) = k\sec^2 kx Therefore, I got: \frac{dy}{dx} = x \sec^2 x However, the answer given is \tan x + x \sec^2 x I can't...
  45. J

    Limits with trig functions manipulations

    I'm pretty useless at trig manipulations, but I'm trying to learn According to my text, the key to limits with trig functions is to get them into this form \lim_ {t\rightarrow 0} \frac{\sin x}{x} = 1 I've been doing alright with that, but I'm stuck on this one \lim_ {t\rightarrow 0}...
  46. W

    How Do You Differentiate y=sec^2(X) Where X is a Polynomial?

    I need a quick check on my math. Derivative of composite function y=sec^2(X) where X is a polynomial. Does it equal: y'=(secXtanX)(secX)(X)(dy/dx)+(secX)(secXtanX)(dy/dx)(X)+(secX)(secX)(dy/dx) I'm a little confused on applying the product rule and the chain rule to this function.
  47. E

    Proving a+b+c=abc with Inverse Trig Functions

    a teacher wrote on the board tan-1(a) + tan-1(b) + tan-1(c) = pi [they are inverse trig functions btw, not the recipricol 1/tanx = cotx] hence prove that a + b + c = abc wow. do you have any idea where i can start? thanks. I've been uttered clueless.
  48. M

    General question about trig functions

    Ok, say you have cos^3(x)... Is this the same as cos(x)^3(where only x is cubed) or (cos(x))^3 (where the whole thing is cubed?
  49. quasar987

    Why are hyperbolic cosine and sine functions named with h in cosh and sinh?

    I just got a clue as to why 0.5(e^x + e^-x) was called "hyperbolic cosine" and 0.5(e^x - e^-x) is called "hyperbolic sine". It is because the "complex version" reads cos(x)=\frac{e^{ix}+e^{-ix}}{2} sin(x)=\frac{e^{ix}-e^{-ix}}{2i} That explains the "cos" and "sin" part in "cosh" and...
  50. A

    Understanding Trig Function Behavior: Period, Amplitude, and Shifts

    Hi just need a little help with the behaviour of this trig function y=-3\cos (2x-\frac {\pi} {6}) +1 I converted the pi over 6 to degrees and got -30 need to state the period, amplitude, max/min values, range, domain, horizontal phase shift, and vertical dispacement. So far after...
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