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. B

    Integral of Logarithms + Trig Functions

    Homework Statement Sec(x)/((ln(tan(x)+sec(x))^1/2) We were instructed to find the integral Homework Equations Here is a link to the wolfram solution, i don't understand the steps they...
  2. Philosophaie

    Trig function of arc trig functions and the reverse

    I know the sin(arccos(x)) = (1-x^2)^0.5 I was wondering what some of the others are: cos(arcsin(X)) tan(arcsin(X)) tan(arccos(x)) sin(arctan(x)) cos(arctan(x)) also the reverse: arcsin(cos(x)) arcsin(tan(X)) arccos(Sin(X)) arccos(tan(X)) arctan(sin(X)) arctan(cos(X))
  3. C

    Finding Limits of Trig Functions

    Homework Statement The first problem I'm having difficulty with is \stackrel{lim}{x\rightarrow0} \frac{sin x}{5x} And the second is: \stackrel{lim}{x\rightarrow0} \frac{sin x(1-cos x)}{2x^{2}} Homework Equations I assume that for the first problem I need to simplify it to the rule...
  4. S

    L'hospital's rule with trig functions

    Homework Statement evaluate lim(x->0) (tan^8(t))dt(between 0 and sin^2x) Homework Equations The Attempt at a Solution [tan^8(sin^2(x))]/sin^18(x) my book says to use l'hospital's rule, so i continued with [8tan^7(sin^2x)*sec^2(sin^2(X))*2sinxcosx] but my book says i should...
  5. Saitama

    Trig Functions Periodicity: Which Function is Not Periodic?

    Homework Statement Which one is not periodic? (a)|sin 3x|+sin2x (b)cos\sqrt{x}+cos2x (c)cos 4x + tan2x (d)cos 2x+sin x Homework Equations The Attempt at a Solution I don't understand how to show whether the functions are periodic or not? :confused:
  6. M

    Using the mean value theorem on trig functions

    Homework Statement let g be a function mapping x to xcosx-sinx. use the mean value theorem to prove that g(x) < 0 for x in (0,pi]Homework Equations well the function is both continuous and differentiable on the interval so that's a start... The Attempt at a Solution basically i thought i'd...
  7. N

    What is the change in horizontal distance from low tide to high tide?

    Homework Statement A cargo ship is tied up at the dock. At low tide, a 12-m long unloading ramp slopes down from the ship to the dock and makes an angle of 30 degrees to the horizontal. At high tide, the ship is closer to the dock, and the unloading ramp makes an angle of 45 degrees t othe...
  8. D

    Finding integrals of the product of trig functions

    Homework Statement I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function. One of them is \intsin(a)*sin(b - a)da Another is \inte(a)*sin(a)da...
  9. T

    Roots of Trigonometric Functions in an Interval

    Homework Statement This isn't really a question on its own, rather a step in the solution to another question: How would I prove that y= A\cos x + B\sin x (A, B arbitrary constants) has at least n zeroes in the interval [\pi , \pi (n+1)] where n\in\mathbb{Z}\;? (I don't need to be too...
  10. M

    Identity Proofs of Inverse Trig Functions

    Homework Statement Prove the Identity (show how the derivatives are the same): arcsin ((x - 1)/(x + 1)) = 2arctan (sqr(x) - pi/2) Homework Equations d/dx (arcsin x) = 1/ sqr(1 - x2) d/dx (arctan x) = 1/ (1 + x2) All my attempts have been messy and it may be because I didn't...
  11. QuarkCharmer

    Differentiating Trig Functions again

    Homework Statement Does this look correct? How do I know when to stop simplifying things? Sometimes it comes out to a nice little expression, and other times it's a long solution. In the latter, I spend too much time trying to simplify it further! Homework Equations The Attempt at a...
  12. G

    Normal polygon area without trig functions

    Here's an interesting problem: How can you find the area of any normal polygon with x sides (or corners) that is inscribed in a circle of radius 1? No trig functions, or things like e or π (Pi), or infinite series, are allowed. If possible, try to avoid summation notation as well, but that might...
  13. A

    Evaluate trig functions at infinity?

    is it meaningful to evaluate cos and sin at infinity? I ask in relation to Fourier integrals... ie does cos(infinity) have a value
  14. C

    How do I graph a horizontal compression with trig functions?

    Homework Statement Y = - sin (2X) Homework Equations The Attempt at a Solution So.. I know how to graph - sin, that's going to be one 1 but reversed. Now my problem is the (2x). How do I graph 2x in the trig graph? How would I go about making a horizontal compression? Thank you
  15. C

    Why does sin80°csc80° equal 1?

    I'm on mobile so I can't use regular symbols. Sin80°csc80° = 1 why does this equal one? Is there multiplication involved here? I know csc = 1/sin
  16. E

    Integration with trig functions

    Homework Statement integral of x^2/sqrt(9-25x^2) Homework Equations The Attempt at a Solution dont know how to type theta so I am using @ so i made x=3sin@ dx=3cos@d@...
  17. D

    Finding the Average Rate of Change for a Trig Function on a Given Interval

    Homework Statement Determine the average rate of change of the function y = 2cos (x - pi/3) + 1 for the following internal: pi/2 < x < 5pi/4 Homework Equations AROC = [ f(x2) - f(x1) ] / x2) - x1 The Attempt at a Solution For an approx. value, I would set the calculator in...
  18. Z

    Are trig functions polynomial fuctions?

    Homework Statement Is 3cos22x + cosx2 - 1 a polynomial function? Homework Equations The Attempt at a Solution
  19. T

    Trig Functions - When wil object be 9cm below 0?

    Trig Functions - "When wil object be 9cm below 0?" Homework Statement Here is the background information: A weight hanging from a spring is set in motion by an upward push. It takes 10 s for the weight to complete one cycle from moving 12 cm above 0, then dropping 12 cm below 0...
  20. A

    Differentiation with trig functions

    Homework Statement An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is given by the following equation, where μ is a constant called the coefficient...
  21. F

    Finding the Value of sin(arctan(3)): Inverse Trig Functions Homework

    Homework Statement Determine sin(arctan(3)) The Attempt at a Solution I do not know how to start this. No special triangles : (.
  22. T

    Determining angles and the 6 trig functions? D:

    Homework Statement The Attempt at a Solution No attempt, don't even know where to start. I get a point... I can draw a triangle out of it. I can figure out the X and the Y. I know the 6 equations... Sin, Cos, Tan, and the opposites Cosecant, Secant, and Cotangent... I know...
  23. T

    General solution of trig functions

    Homework Statement Find the general solution of: a) sin x = 1/\sqrt{2} b) cos x = 0.5 The Attempt at a Solution a) x = asin(1/\sqrt{2}) x = (π/4) + 2nπ b) x = acos(0.5) x = π/3 + 2nπ Basically, my strategy was to solve for the basic angle, and then add multiples of the...
  24. C

    Trig functions within Trig functions

    Homework Statement I'm given the problem: \int\frac{\sqrt{y^{2}-25}}{y} dy Homework Equations I set y = 5sec(u), and solve for the subsitution. The Attempt at a Solution At the culmination of my solution, I achieve: 5tan(u) - 5u + C Here is my dilemma, u stands for...
  25. C

    Integrals involving trig functions

    Homework Statement I need help evaluating the integral Cotx3/10 I factored out the 1/10 from the integral and am just left with (1/10)*Cotx3 from here i do not really know what to do. I rewrote it in terms of sine and cosine to get (1/10)*(Cosx3/Sinx3)dx I multiply the integral...
  26. M

    Inverse trig functions with tan-1

    Homework Statement sin(tan-1(x)) Homework Equations The Attempt at a Solution y=tan-1(x) tan(y)=x sec2(y)= 1+tan2(y) sec(y)=\sqrt{1+x^2} This is where I'm getting stuck. I know that I have to say that the sin(y)= whatever, but I'm not sure how to tie the sin sec and tan...
  27. J

    Integration: completing the square and inverse trig functions

    Homework Statement Find \int(x+2)dx/sqrt(3+2x-x2) Homework Equations \intdu/sqrt(a2 - u2) = sin-1u/aThe Attempt at a Solution I began by completing the square: 3+2x-x2 = 4 -(x2-2x+1) So, 4-(x-1)2 = a2-u2 and a=2 and u=(x-1) Further, since x=(u+1), dx=du and (x+2)=(u+3) Substituting, I...
  28. G

    Differentiate e^x and Trig Functions

    Homework Statement Differentiate e^x * cotx / 5sqrtx^2 [Sorry for not using the formatting things. They didn't seem to be working for me, and this is urgent!] Homework Equations The quotient rule seems like that's the way to go... The Attempt at a Solution At first I tried using...
  29. M

    Inverse trig functions and pythagorean identity

    Hi. I'm having trouble trying to understand the relationship between inverse trig functions, especially arcsin x, and pythagorean identity. I know that because cosx=sqrt(1-(sinx)^2), derivative of arcsin x is 1/(cos(arcsin x)) = 1/(sqrt(1-(sinx)^2)arcsinx)) = 1/(sqrt(1-x^2). But how does...
  30. Somefantastik

    How Can We Determine the Max Min of Trig Functions in the Range (0, pi)?

    x in (0,pi]; f(x) = sin(x)-x2; f'(x) = cos(x) - 2x; f'(x) = 0 ==> cos(x) - 2x = 0; since |cos(x)| ≤ 1, cos(x) - 2x ≤ 1 - 2x; Now 1-2x = 0 <==> x = 1/2; f'(1/4) = cos(1/4) - 2*(1/4) > 0 and f'(3/4) = cos(3/4) - 2*(3/4) < 0; ==> x = 1/2 is maximum and f'(x) ≤ 1/2; Is my...
  31. S

    Finding intervals of trig functions

    Homework Statement solve the equation for x in the interval 0<=x<=2pi 4cos(2x)+sin(x)=4 The Attempt at a Solution I don't understand what the question is asking me to do? Where do I start and how can this equation be made into an appropriate equation so i can answer the question?
  32. D

    Linear Combinations of Trig Functions - Finding Roots

    Hi there I was wondering if there is a simple way to solve for the roots of a complicated summation of trig functions that can't be combined with any simple identities. I have an equation of the form: 0 = sin(8x-arctan(4/3))+3.2sin(16x+pi/2) where the two sines have different amplitudes...
  33. M

    Understanding the Power Reducing Identity in Trigonometric Functions

    Homework Statement See attached image Homework Equations Power reducing Identy: cos^u = (1+Cos2u / 2) The Attempt at a Solution This is a problem that was done as an example (I know it is not complete it is missing the last integration step) but this is where I am getting stuck my...
  34. H

    Convergence of a Series with Trig Functions

    Homework Statement *The sum from n = 1 to infinity* (cos(n*pi))^2/n*pi where n is the variable. Homework Equations All forms of convergence tests (i.e. Ratio Test, Integral Test, Alternating Series test etc.) The Attempt at a Solution I am having trouble figuring out which route...
  35. S

    Inverse Trig Functions: Evaluating Expressions in Radians

    Evaluate the following expressions. Your answer must be in radians. a) arctan(-(sqrt3)/3) b) arctan((sqrt3)/3) c) arctan(-sqrt3) What I got are: a) 5pi/6 b) pi/6 c) 2pi/3 Do I got these answers right?
  36. L

    Optimize Derivative of Trig Functions Grade 11 Math

    Homework Statement there's a picture of the question... from my textbook http://photos-h.ak.fbcdn.net/hphotos..._1385551_n.jpg thers a diagram image of the problem too to help understand Homework Equations well its a word problem, i used cosine rule at beginining and then...
  37. L

    Derivative optimization trig functions, give it a try please grade 11 math

    could someone please try and solve this? and explanation would be greatly appreciated too ! this was one of the homework questions, but i didnt really understand. the teacher explained it again to the class partly, but didnt understand a part of it so we didnt continue... maybe one of you guys...
  38. T

    Exact Values of Trig Functions at 1/3pi and 1/6pi with Symmetry - Homework Help

    Homework Statement (In radians I assume) Using the exact values of sin, cos, tan of 1/3 pi and 1/6 pi, and the symmetry of the graphs of sin, cos and tan, find the exact values of sin(-1/6pi), cos (5/3pi) and tan(4/3pi). Homework Equations cos(x)=sin(x+pi/2) sin(x) = sin(x+2pi)...
  39. Char. Limit

    Exploring the Astonishing Properties of Trig Functions

    ...are fascinating. At least I think so. Sine and cosine are the additive inverses of their respective second derivatives, for example. Astonishing! Are there any other startling properties of trig functions (not inverse trigs) that would just blow my mind? Somewhere in the beautiful...
  40. T

    How to Integrate cosx/((sinx)^2 + 1) with Respect to x

    how do i integrate -- cosx/((sinx)^2 + 1) with respect to x thanks for your help
  41. L

    Need Help Understanding Trig Functions

    All my life I've understood Trig Functions as ratios of sides of a triangles. With this understanding I have been able to get the ratios for simple triangles like 30, 45, etc... since my teachers made me memorize the sides of it... But now I'd like to know how would you find the ratio of a...
  42. M

    Limit of Trig Function as x Approaches 0: Is the Answer 0/0?

    Homework Statement lim (cos x - 1) / (sin^2 x + x^3) as x approaches 0. Homework Equations sinx/x = 1 The Attempt at a Solution I get 0/0. Is that the answer?
  43. C

    Finding the Second Derivative of a Trigonometric Function

    Homework Statement y' = csc2(\vartheta / 2 ) Find y" The Attempt at a Solution So far, i have 2 csc (\vartheta/2) *csc(\vartheta/2)cot(\vartheta/2) * 1/2 but I'm wondering, how do you take the derivative of a half angle identity? or does it just simplify down to csc2(\vartheta...
  44. Z

    Power series of inverse trig functions

    How do you find the power series for inverse trig functions? Can I find the power series for arcsin by manipulating the power series for sin? Thanks!
  45. N

    Graphing trig functions without calculus

    Homework Statement I'm in a first-year analysis course, and this question was given by my prof. as practice for her midterm test. "Sketch the graph of the function \begin{equation*} f(x) = \text{sin} 2x + \sqrt{3} \text{cos} 2x \end{equation*} Determine the amplitude, the frequency and...
  46. Z

    Partial Fractions (with trig functions)

    Homework Statement Integral(sinx(x)dx/(cos^2(x)+cos(x)-2) Homework Equations The Attempt at a Solution What I tried to do first was factor the denominator, so i got (cos(x)-1)(cos(x)+2) from there, I set up my partial fractions equation trying to solve B(cos(x)-1) + A(cos(x)+2) =...
  47. P

    Limits of Trigonometric Functions

    Homework Statement lim as x approaches 2 (cos(pi/x))/(x-2) lim as x approaches pi/4 (tan(x)-1)/(x-(pi/4)) Homework Equations equations above The Attempt at a Solution for the first limit, i tried substituting t = (pi/2)-(pi/x) but i got stuck i have no idea how to do the...
  48. W

    Contour Integration with Trig Functions

    Homework Statement My question has two parts. The first part is the solution of the following integral: \int^{\infty}_{-\infty}\frac{cos\:x\:dx}{1+x^{2}} They give the answer as being \frac{\pi}{e} This is actually an example problem in the book, but I don't understand how...
  49. F

    Calculus with inverse trig functions

    Homework Statement Evaluate the integral of (1/Sqrt(5x-x^2)) Homework Equations [d/dx]{arcsin(x)}=(du/dx)/sqrt(1-x^2) The Attempt at a Solution arcsine(2x-5)/5 I did end up getting the right answer, but have no idea how I got there.
  50. L

    Natural Domain of trig functions

    Homework Statement Find all the natural domain of the function algebraically, and confirm that your result is consistent with the graph produced by your graphing utility. h(x) = 3/2-cosx Homework Equations (a) h(x) = 3/2-cosx (b)x2-1/(x+1) The Attempt at a Solution Do I need to...
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