Trigonometric functions Definition and 163 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. T

    Second Derivative of Trigonometric Functions

    Homework Statement Find the second derivative (y") of y=xtanx. The attempt at a solution I got the first derivative (y') y=xtanx y'=x(secx)+tanx I started the second derivative and got stuck y"=xsec^2x+tanx
  2. P

    Period of non trigonometric functions

    is there any definite way of finding the period of non trigonometric functions? can we use f(x+t)=f(x) and solve for t from this equation?
  3. Mentallic

    Trigonometric functions: express sin(x) in terms of tan(x)

    Homework Statement I want to express sin(x) in terms of tan(x).Homework Equations tan(x)=sin(x)/cos(x) 1+tan2(x)=sec2(x)The Attempt at a Solution sin(x)=cos(x)tan(x) At this point I realize this is assuming x\neq \pi/2+k\pi cos^2(x)=\frac{1}{1+tan^2(x)} therefore...
  4. S

    Periods of Powers of Trigonometric Functions

    Homework Statement Is there a way to determine the period of a function like f(x) = a*sin(b*x)^2 + c*cos(d*x)^2 + e*sin(f*x) + g*cos(h*x)? Homework Equations The Attempt at a Solution I know how to find the periods of sines, cosines, and arbitrary sums of the two, but the...
  5. J

    Integrals of trigonometric functions over [o,2pi]

    Homework Statement ∫dθ/(1+βcosθ)^2 ; -1<β<1 θ=0 to 2pi Homework Equations The Attempt at a Solution attempt solution: 1) make substitution: dθ=dz/iz Z=e^iθ cosθ=1/2(Z+1/z) 2) substitute: 1/i*dz/(β+Z(1+(β^2)/2)+((3βZ^2)/2)+((β^2)Z^3)/4)+((β^2))/4Z) 3) Next ...
  6. G

    Limits of Trigonometric Functions.

    Limits of Trigonometric Functions. URGENT! Homework Statement Evaluate stackrel{lim}{x \rightarrow0}[/tex] [sin(\frac{2e}{x3}) \bullet (arctanx)] Homework Equations All I know is that the equation stackrel{lim}{x \rightarrow0}[/tex] \frac{sin x}{x} = 1 might be helpful, but I'm not sure...
  7. T

    Derivatives of trigonometric functions

    Homework Statement Find the constants A and B such that the function y = Asinx + Bcosx satisfies the differential equation y'' + y' - 2y = sinx Homework Equations None The Attempt at a Solution My attempt: y = Asin x + Bcosx y' = Acosx - Bsinx y'' = - Asin x - Bcosx y''...
  8. M

    On Inverse Trigonometric Functions

    Hi, can you help me solve these three questions? Please show each step. Thanks. 1. solve for x: arcsin(6x-pi)=1/8 2. Find an equation of the tangent line to the graph of y = arcsin (6x) at the point ((1/(6sqrt2),(pi/4)) 3. Find the indefinite integral of 1/sqrt(81-100x^2)
  9. J

    Integral Involving Trigonometric Functions with Varying Arguments

    Homework Statement I'm in an Intermediate Mechanics course right now, and while the Physics itself isn't giving me too much trouble, I am lagging behind in the Math department. I am trying to solve the integral: \int cos(\omega t) sin(\omega t - \delta) dt Homework Equations...
  10. morrobay

    Hyperbolic trigonometric functions in terms of e ?

    cosh x= e^x+e^-x/2 sinh x= e^x-e^-x/2 Can someone explain why the hyperbolic trigonometric functions are defined in terms of the natural exponential function, e^x ?
  11. I

    Single Trigonometric Functions ( trig identities)

    Homework Statement Cos^2x-Sin^2x/2 SinxCosx The Attempt at a Solution I changed cos^2x to 1- sin^2x which then the equation was 1- s sin^2x/2snxcosx and i have no idea how to make this a single trig. function
  12. Z

    Composition of trigonometric functions, mean value theorem

    Homework Statement how to show using MVT that cos(cos x) is a contraction. Homework Equations | d/dx (cos(cos x)) | = | sin(cos x) sin(x) | < sin 1 < 1 The Attempt at a Solution Using that relation, the original problem is easily solved. My question is, how do we know: |...
  13. D

    Integral of trigonometric functions

    Homework Statement How can I integrate this: \int sin (nt) sin (n \pi t) dt This actually in the Fourier series.
  14. J

    Limit with trigonometric functions

    Homework Statement True or false? lim Tanx/(1-Cosx) = lim Sec2x/Sinx = Infinity (limits are as x approaches \pi from the left) The Attempt at a Solution I tried just plugging \pi into x in the first limit, and I ended up getting 0/2, which exists but is just 0. So...
  15. Z

    Link between Pascal's triangle and integrals of trigonometric functions

    My lecturer keeps simplifying trigonometric integrals in one line such as \int^{2\pi}_{0}sin^{4}(t)dt=\frac{3\pi}{4} and writes pascals triangle next to it. Just wondering what's the link between them? I'm sure it's obvious and easy, I'd just like to have an fast way of dealing with these
  16. R

    Integrals of Trigonometric Functions

    I have three problems that I'm having a hard time with. I'd appreciate any help with any of the three problems. \int((cos(x))^6)dx AND \int(x^3 * sqrt(x^2 - 1) AND Solve for y (separation of variables): dy/dx = ((2y +...
  17. R

    Integrals of Trigonometric Functions

    Homework Statement I have three problems that I'm having a hard time with. I'd appreciate any help with any of the three problems. \int((cos(x))^6)dx AND \int(x^3 * sqrt(x^2 - 1) AND Solve for y (separation of...
  18. J

    Prove 1+1=2 using trigonometric functions

    Prove 1+1=2 using trigonometric functions
  19. ShayanJ

    The period of trigonometric functions

    Hi everyone Could you give me a way to calculate the period of every trigonometric functions? thanks
  20. R

    Graphs of Reciprocal Trigonometric Functions

    Homework Statement A road is built up the slope of a hill with a height of h meters and an angle of inclination of x. The length of the road is d. a) Sketch a diagram of this situation. Label all quantities. b) Show that the length of the road is represented by the relation d = h csc x. c)...
  21. P

    Manually Graphing Trigonometric Functions - Turning Points, Extrema

    Homework Statement Two curves y_1=(\frac{20}{x^2})\sin(\frac{10}{x}) and y_2=5\cos x intersect in three points in the interval (1,3). Draw the graphs, compute the minimum and maximum points as well as the turning points and show the points of intersection. 2. The attempt at a...
  22. A

    Help needed with Verifying Trigonometric Functions

    Homework Statement \frac{secx-cscx}{secx+cscx}=\frac{tanx-1}{tanx+1} \frac{(tan^{2}x - cot^{2}x)}{(tanx + cotx)}= (tanx - cotx) tan^{2}2x+sin^{2}2x+cos^{2}2x=sec^{2}2x cot^{2}2x+cos^{2}2x+sin^{2}2x=csc^{2}2x The Attempt at a Solution I've tried many times in my notebook and I'm...
  23. R

    Finding the limit of trigonometric functions?

    Im having trouble with this problem here: x-->0 tan6x / sin2x So far I only have: (sin6x / cos6x) / (2sinx cosx) What would be the next step? Multiply the reciprocal? Also, could somebody tell me if I am doing this problem right? x--> 0 (sin (cosx)) / secx sin (cosx) / 1/cosx...
  24. H

    Integrating Trigonometric Functions with Irrational Exponents

    Homework Statement The integral from 0 to pi/2 of 1/(1 + (tanx)^sqrt2) dx. Homework Equations trig identities? The Attempt at a Solution I tried some substitutions but it just made the problem more complicated. I also multiplied by (tanx)^sqrt2 in the numerator and denominator in...
  25. N

    Trigonometric functions and the unit circle

    Homework Statement Hi all. Today I had to solve: \cos \theta = -1/2. What I did was to look in a table to find that \theta = 2\pi/3 \quad \text{and}\quad \theta = 4\pi/3. My question is what is the general strategy when I wish to write this as a a function of an integer n? Is there even a...
  26. M

    Differentiation of Trigonometric functions

    Homework Statement Let y = tan2(3x-2) Find dy/dx The solution is: 2*tan(3x-2)*sec(3x-2)*3 = 6*tan(3x-2)*sec2(3x-2) Why is it not: 6*tan(3x-2)*sec(3x-2) I am thinking: y = (tan(3x-2))2 take the power 2 down,multiply with the parentes multiply with the defferentiated parentes...
  27. F

    Trigonometric functions - angular measures and tangent curves

    I encountered a few problems for a few questions while doing my homework. 1. Angular measure problem: A Ferris wheel with a radius of 25.3m makes 2 rotations every minute. a) Find the average angular speed of the Ferris wheel in radians per second. b) How far does a rider travel if the ride...
  28. K

    Unit Circle Trigonometric Functions

    Homework Statement I'm trying to do a few problems that ask me to "find the point (x,y) on the unit circle that corresponds to the real number t." Examples of these problems are: t = pi / 4 t = 7pi / 6 t = 4pi / 3 etc etc Homework Equations The Attempt at a Solution I...
  29. L

    Periodicity of Inverse Trigonometric Functions

    Homework Statement My problem from before has been more or less resolved, but now I have a new, bigger problem. I need to figure out how to find recuring values for trig functions. I'm having a hard time figuring out how to 1. Get the equations associated with a given value for the trig...
  30. redtree

    Why Wave Functions Use Complex Exponentials vs Trigonometric Functions

    Why are wave functions, e.g., Schrodinger's, based on the complex exponential function (e^{}ix) and not trigonometric functions (sine or cosine)? See Euler's formula for their relationship: http://en.wikipedia.org/wiki/Euler%27s_formula Furthermore, by using the complex exponential...
  31. L

    Derivatives of trigonometric functions

    Hi, I'm working with finding the derivatives of trigonometric functions but I'm not confidant with some of my answers. if someone would go over these derivatives i would appreciate it. thanks in advance! determine \frac{dy}{dx} . do not simplify. question 1 y = sec \sqrt[3]{x} my...
  32. L

    Limits of trigonometric functions

    Hello, i'm having some trouble with evaluating limits if anyone could help me out a bit i would appreciate it. thanks in advance evaluate the following limits: Question 1: lim x -> 0 \frac{2tan^{2}x}{x^{2}} my answer: u = x^{2} as x -> 0 u-> 0 = lim u->0...
  33. F

    Derivative of trigonometric functions

    y = sin( 4 x ) cos( 3 x ) f(x) = sin4x g(x) = cos3x f'(x) = cos4x g'(x) = -sin3x And by using the product rule, i'll get: cos4x(cos3x) - sin4x(sin3x) Is the answer correct or can be simplify again??
  34. B

    Derivatives of trigonometric functions

    Homework Statement I'm learning now about darivative all by my self (without a teacher) and I'm not sure about this development Homework Equations The Attempt at a Solution Y=senx y+{\Delta}Y=sen(x+{\Delta}Y) {\Delta}Y=senx*cos{\Delta}x+ sen{\Delta}x*cosx What about...
  35. J

    Limits of trigonometric functions

    Homework Statement lim x --->0 [tan(2+x)^3 - tan8]/x Homework Equations f'(a)=lim h-->0 f(a+h)-f(a)/h The Attempt at a Solution should i differentiate first? am i allowed to do that?
  36. D

    Derivatives of Trigonometric Functions

    [SOLVED] Derivatives of Trigonometric Functions I need to find the critical numbers of this function: y = cos x - sin x where -pi <= x <= pi I found the derivative as: dy/dx = -(sin x + cos x) But when I equate dy/dx to zero, I get: sin x + cos x = 0...where do I go from here?
  37. W

    Trigonometric Functions And Identities

    Hi there, I am struggling to solve for x in the following problem:- Find all values of x in the interval 0<= x <= 360 for which: tan^2(x) = 5sec(x) - 2 I have used the identity tan^2(x) + 1 = sec^2(x) to get: sec^2(x) - 1 = 5sec(x) - 2 and rearranged to get sec^2(x) - 5sec(x) +...
  38. O

    Limits of trigonometric functions

    Why do some problems return the wrong answer while others do not on the ti-89. For example: \[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\] Shows up wrong (shows up as pi over 180). But \[ \lim_{x \to 0} \frac{\sin x(1 - \cos x)}{2x^2}\] does not?
  39. T

    Solving an Equation Involving Trigonometric Functions

    Homework Statement (1-cos^2x)(1+tan^2x) = tan^2xHomework Equations N/AThe Attempt at a Solution (1-cos^2x)(1+tan^2x) = tan^2x L.S. = (sin^2x)(1+sin^2x/cos^2x) = sin^2x+(sin^4x/cos^2x) Now, I get a common denominator, but it's not doing anything for me. Did I do the right thing in converting...
  40. L

    Derivatives of Trigonometric Functions

    1.A kite 40 m above the ground moves horizontally at the rate of 3 m/s. At what rate is the angle between the string and the horizontal decreasing when 80 m of string has been let out. Answer is 0.02 m/s 2. What I did was: -Drew a triangle as prescribed above -I found the unknown...
  41. A

    Series with Hyperbolic and Trigonometric functions

    Homework Statement Determine whether the series converges and diverges. \sum_{n=3}^{\infty}\ln \left(\frac{\cosh \frac{\pi}{n}}{\cos \frac{\pi}{n}}\right) The Attempt at a Solution \sum_{n=3}^{\infty}\ln...
  42. H

    Derivatives of Trigonometric Functions

    Homework Statement Find d^2y/dx^2. y = x cos x The Attempt at a Solution I've been doing derivatives recently and now got into doing them with trig functions. I thought it was, y = x cos x = -xsinx = -xcosx but that is the derivative of the derivative. The problem...
  43. A

    Derivative trigonometric functions help

    Derivative trigonometric functions Homework Statement Find d^{2}x/dt^{2} as a function of x if dx/dt=xsinx Homework Equations The Attempt at a Solution I tried to solve the problem by taking a second derivative of dx/dt=xsinx but I was not sure how to start. Also, it is...
  44. T

    Inequalities with trigonometric functions

    Homework Statement Three functions are defined as follows: f:x> cos x for the domain 0< (or equal to) x < (or equal to) 180 g:x> sin x for the domain 0< (or equal to) x < (or equal to) 90 h:x>tan x for the domain p< (or equal to) x < (or equal to) q Find the range of f. -1<(or...
  45. U

    A little help with integration and trigonometric functions.

    I was doing my exam today and ran into a couple problems. First one: how do you differentiate \tan^2? I converted it into \sec^2 - 1 and used the u/v = (u`v - v`u)/v^2 method, but I would like somebody clever to do it for me, just to be sure, please. Homework Statement Another problem. Rate of...
  46. H_man

    Error Propagation in Trigonometric Functions

    Homework Statement I can't seem to find online how to calculate the error propogated by trigonometric functions. That is, I know the uncertainty in \theta but am not sure how to deal with it when I apply the tan function. I am quite okay with how to deal with all the basic...
  47. A

    Trigonometric functions of the unit cirlce

    I just came across a problem that wants you to solve for csc of an angle in radians... However, I'm confused about the answer given. Here is the problem: Find the csc when t=-2pi/3, which is equivalent to -120 degrees right? I got -2sqrt.(3)/3 but the answer in the back is -2sqrt.(3)/2...
  48. B

    Wave motion is expressed with trigonometric functions

    I was wondering if someone would be able to help me with the following questions: -A progessive wave has amplitude 0.40m and wave length 2.0m. At a given times the displacement y=0 at x=0. Calulate the displacement at (a)t=5sec (b) t=0.8sec -A progessive wave has amplitude 2.5m and a time...
  49. R

    Arccos(x+y)=? addition theorems for inverse trigonometric functions?

    Hi. Are there any addition theorems for inverse trigonometric functions? Like arccos(x+y)=? or something... I was wondering about this when I tried to find the derivative of f(x)=arccos(x) by setting f'(x)=\frac{\arccos(x+\Delta x)-\arccos(x)}{\Delta x}
  50. M

    Trigonometric functions and radians

    Solve the following equation giving values from -\pi to \pi: cos (2v - \frac{\pi}{3}) = \cos v Here is my attempt to solve it. As the cosine of the two is the same, the angles should also be the same leaving 2v - \frac{\pi}{3} = v + 2 \pi n Then if I move the right over to the left, I get...
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