What is Trig functions: Definition and 218 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. R

    Finding Inverse Trig Functions: Converting Between Point and Radian Measures

    Homework Statement Cot^-1(-sqrt(3)) and CSC(arccos(3/5) Homework Equations The Attempt at a Solution I know this looks like a trig problem, but I'm in calc, just wasn't sure where to put this. I have the solution to both problems, my biggest issue here is that I do not know of...
  2. D

    Limits of trig functions

    1. what would be the limit?? without using the L'Hopital's rule lim_(x-0) (sin(3 x^2))/(8 x) the limit of sin(3x^2) divided by 8x as x approaches zero 2. Limits of trignometric functions 3. The Attempt at a Solution I tried factoring out the 1/8, but...
  3. M

    Integrating Sin^2 (2x) Without Prefix

    ## \frac {1}{4} \int sin^2 (2x)dx = I = \frac {1}{4} [- \frac {1}{2} sin(2x)cos(2x) + \int cos^2 (2x)dx]## when ##u = sin(2x), dv = sin(2x)dx, v= - \frac {cos(2x)}{2}## and ##du = 2cos(2x)dx## Now simplifying ##\int cos^2 (2x)dx## you get ## x - \int sin^2 (2x)dx = x - I## Then, ## I =...
  4. B

    General solution of trig functions

    Hello, Homework Statement find the general solution to cos3θ = sin2θ Homework Equations The Attempt at a Solution I know that sinθ = cos(π/2 - θ) but I am unsure of how to apply this when I have sin2θ. Do I say that sin2θ = cos2(π/2 - θ)? I think not because when I do...
  5. Lebombo

    A curve that intersects itself at some point w/o trig functions

    In order for an equation to be a function, it has to pass the vertical line test. A circle is not a function because it does not pass the vertical line test. A curve containing a loop does not pass the vertical line test and to me that means it is not a function. However, if I am given...
  6. T

    Manipulating Trig Functions

    Homework Statement Write sin(7t)-sin(6t) as a product of two trig. functions. Homework Equations e^(ix)=cos(x)+isin(x) sin(2x)=2cos(x)sin(x) cos(2x)=cos^2(x)-sin^2(x) The Attempt at a Solution I do not really know how to approach this. I have tried using the sin(2x) identity...
  7. 462chevelle

    Two Solutions for Trig Functions: An Intuitive Way

    when working with trig functions. is there a trick to knowing if there are 2 solutions while filling out a triangle without memorizing your sines and cosines. or do I need to be subtracting all numbers I get by 180 then sin/cos them to see if they have the same number? doing law of sines right...
  8. C

    Solving systems of equations with trig functions

    I've stumbled upon a system of equations that involves trig functions... 100cos(θ) + 200cos(ω) = 250 100sin(θ) + 200sin(ω) = 0 How do you go about solving a system like this? It's nonlinear, so linear algebra won't work...
  9. P

    Unboundness and periodicity for complex trig functions

    Hi I just found out that cos(z) and sin(z) are unbounded and tend to ∞ which I find strange ! But the part I'm struggling with is that I can't reconcile that fact with the fact that they both have a period of 2pi. Surely that means that each value in the range 0-2pi is repeated in the range...
  10. M

    Lim of trig functions. Does it exist?

    1. Does the limit exist of the following: lim as x→ 1- ((cos^-1(x))/(1-x)) 2. Homework Equations : 3. The Attempt at a Solution : lim as x→ 1- ((cos^-1(x))/(1-x)) = lim as x→ 1- (cos^-1(x))/ lim as x→ 1-(1-x) Let y = 1-x lim as y→0 (cos^-1(1-y)) / lim as y→0 (y) =...
  11. B

    An integral with exponential, and trig functions within trig functions

    I'm working with the integral from 0 to infinity of t^(x-1)e^(-atcos(b))cos(atsin(b)) with respect to t. specifically, I'm asked to solve in terms of the gamma function. my question is more of what general technique i should use. all I've been able to do so far is beat it to death using...
  12. G

    Understanding the Inequality in Trigonometric Function Analysis

    Homework Statement This isn't really a homework question, just working through Rudin and got caught up on something. C(x) and S(x) refer to cos(x) and sin(x) respectively. Here is the section in question: http://grab.by/mSo8 Homework Equations The Attempt at a Solution Well the part I'm...
  13. M

    Finding derivatives of inverse trig functions using logarithms

    For some polynomial functions it is useful to logarithmize both sides of the eq. First. How can this be applied for inverse trig functions? Is it even possible?
  14. D

    Finding the Limit of Trig Functions: 2x/sin3x, x->0

    Homework Statement lim 2x/(sin3x) x-> 0Homework Equations lim sinx/x = 1 x->0 The Attempt at a Solution is it correct to say the following: lim 2/3 (sinx/x) x-> 0 lim 2/3 (1) x-> 0 Answer: lim 2x/sin3x = 2/3 x-> 0 Because it's on the book: cos 2x(2)/3...
  15. Petrus

    MHB Limit of Ratio of Difference of Trig Functions at $\pi/4$

    $$\lim_{x \to \pi/4} \frac{1-\tan(x)}{\sin(x)-\cos(x)}$$ progress: I start with rewriting $\tan(x)=\frac{\sin(x)}{\cos(x)}$
  16. W

    Limit (L'hopitals) trig functions

    Homework Statement use l'hopital's to evaluate the limit. Homework Equations limit (∅->0) ∅-3sin∅cos ∅ -------------------- tan∅- ∅ The Attempt at a Solution i take the derivatives of the top and bottom, and use...
  17. L

    Derivative of Trig functions problem.

    Homework Statement If x = asecθ, y =btanθ show that dy/dx = (b/a) cosecθ and d2y/dx2 = (-b/a2)cot^3θ The attempt at a solution I got the 1st part dy/dx = (dy/dθ) * (dθ/dx) = bsec^2θ x 1/(secθtanθ)= (b/a) cosecθ Now I tried differentiating a 2nd time and I don't get...
  18. D

    Derivative of Trig Functions: Solving for f'(x) and f(∏/6)

    Homework Statement x3 - sin 2x Find f'(∏/6)The Attempt at a Solution f'(x) = 3x2 - 2 cos 2x f(∏/6) = 2700 - 2 [ (√3/2) ] ---> from 2 [ cos(∏/6)] answer: 2700 - √3 My book has the answer as (∏2 - 12)/12
  19. D

    Derivative of trig functions

    Homework Statement 1. f(x) = 5 sin (8∏x) 2. g(x) = 4∏ [ cos (3∏x) sin (3∏x)] 3. h(x) = cos [sec (5∏x)] 4. Sketch the graph of each function on the indicated interval, making use of relative extrema and points of inflection. f(x) = 2sinx + sin2x ; [0,2∏] The Attempt at a...
  20. D

    Derivative of trig functions

    Homework Statement y = √sinx The Attempt at a Solution y' = [(sinx)1/2]' y' = 1/2 (sinx)-1/2 (sinx)' y' = 1/2 cosx (sinx)-1/2 However book says the answer should be: 1/2 cotx (√sinx)
  21. TalkOrigin

    Help with Trig Functions - min/max values and least + values of x

    Hi, So I'm stuck on a part of trig which I can't seem to wrap my head around. I'm self teaching so no teacher to ask unfortunately. The question(s) come in the form: "Find the the max and min value of each of the following functions. In each case, give the least positive values of x at which...
  22. R

    Error propogation in trig functions

    Homework Statement I have to use this in my calculation \theta=\tan^{-1}\left(\frac{19 \pm 1}{47 \pm 1}\right) where both are in mm. How would I get this into \theta\pm \text{error}? Homework Equations shown above The Attempt at a Solution looked through my lab manual, it wasn't...
  23. X

    Breaking up two trig functions

    Homework Statement attachment Homework Equations The Attempt at a Solution ok can someone explain to me how this is done. I have no idea how in hell it got broken into those two.
  24. M

    Trig functions cross multiplying?

    trig functions cross multiplying?? Homework Statement sinx/cosx - 2sinxcosx/1Homework Equations none??The Attempt at a Solution when I cross multiply, should it be sinx-2sinxcosx/cosx or sinx-2sinxcos^2x/cosx ?? here's a pic: http://tinypic.com/r/24fgvmv/6
  25. V

    Equation help with trig functions

    I am currently working a physics problem and I have run into some math that I don't understand. y = 4.0m + 4.0m(sin theta) = 4.0m(1+sin(theta)) In the problem I am trying to find a specific height at a certain angle (pendulum problem). I have found some help online that walks me through...
  26. V

    Determinant of Matrix Involving trig Functions

    Homework Statement Find the determinant of the matrix {{cos 25°, sin° 65}, {sin 120°, cos 390°}} (sorry, can't latex). {cos 25°, sin° 65} is first row and {sin 120°, cos 390°} is the second one. Homework Equations cos(a + b) = (cos a)(cos b) - (sin a) (sin b) The Attempt at a...
  27. C

    Laplace Transform of squared trig functions help?

    now say we have cos^2(3t), how would you go about computing it with the 3t? i can manage cos^2(t) but I'm not sure how to take it that one step further in the link below is what I've managed so far.. SOLVEDI worked it out. If anyone's interested in the future, Just start it off as cos^2(t)...
  28. A

    Solving for v(f) using trig functions.

    Hi everyone, a classmate and I are studying for a test and have been trying to work out the following problem for the past hour and a half with absolutely no progress. Please point us in the right direction :) Homework Statement Someone at a third floor window (12m above ground) hurls a ball...
  29. T

    Trig functions, finding co-ordinates

    Homework Statement I have a graph with the functions f(x)=sin2x and g(x)=cosx. The 2 graphs intersect at point B. They want me to find the co-ordinates of B. Homework Equations The Attempt at a Solution Must I equate the two graphs? sin2x = cosx 2x = 90-x, 3x = 90, x=30...
  30. M

    Can you tell me why my trig functions aren't working?

    I am given two sides of a triangle and the angle in/between them: 9 in/s and 4.5 in/s at 50 degrees. I am using the Law of cosines to get the third side which is 7.013 in/s. I then used the law of sine to find the two remaining angles. I have continually gotten 79.4 for one angle and 29.4...
  31. N

    MHB Reciprocal trig functions

    After a long summer, I finding my new C3 homework a bit tricky, so any help would be great! Here is the question: sec(θ-150 degrees)=4 (solving for theta is greater than or equal to -180, but less than or equal to 180) So I know that sec is the reciprocal of cos so I changed the equation to...
  32. K

    Continuity of piecewise defined trig functions

    Homework Statement Define functions f and g on [-1,1] by f(x) = xcos(1/x) if x≠0 and 0 if x = 0 g(x)= cos(1/x) if x≠0 and 0 if x = 0 (These are piecewise defined. I don't know how to type them in here.) Prove that f is continuous at 0 and that g is not continuous at 0. Explain why...
  33. V

    Help with inverse trig functions

    Here is my problem: cot(arcsin(x)) my awnser: cot= x/(1-x)^1/2 The online program were suppose to use says I am wrong but I am not sure what I did wrong.
  34. S

    Integration involving trig functions and various powers of X

    ∫[6x^6 sin (9x)]/[1+x^10] * dx I've set u =x^6 du=6x^5*dx dx=du/6x^5 ∫[6x^6 sin (9x)]/[1+x^10] * (du/6x^5) = ∫[x*sin(9x)*du]/1+x^10. Can someone help me figure out the next step? I'm thinking of putting a constant out in front, so I can use 2du for (x^10)
  35. S

    Integration Using U-Substitution involving Trig Functions and Identities

    1.) ∫[(7 sin (x))/[1+cos^2(x)]] * dx 2.) I'm looking at the trig identity sin^2 x+cos^2 x=1, and am wondering if I could use that in solving the problem. Or should I use u=sin x, then du= cos x, then plug those in? 3.) so I thought maybe it would be easier to separate the two...
  36. L

    Formal definition of derivative: trig vs non trig functions

    for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 \frac{f(x+Δx)-f(x)}{Δx} this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and verticle is y. But sinx is a trig function and trig functions...
  37. P

    Evaluating Limits with trig functions

    Homework Statement lim x-->0 sin(pi/x) sqrt(x^3+x^2) The Attempt at a Solution I was having trouble evaluating the above limit. Do I start by isolating x? For some reason, when it comes to trig functions such as this, I'm not sure how to simplify it. Also, what material would I have to...
  38. S

    MHB Evaluating Integrals Involving Trig Functions

    Evaluate: 1. $\displaystyle \int_0^{\displaystyle 2\pi} \frac{x \sin^{2n}(x)}{\sin^{2n}(x)+\cos^{2n}(x)}dx$, $n>0$ 2. $\displaystyle \int_0^{ \displaystyle \pi \over \displaystyle 2} \frac{x \sin x \cos x}{\sin^{4}(x)+\cos^{4}(x)}dx$
  39. H

    MHB Factoring exponents from trig functions

    I tend to forget some of the trigonometric functions and someone showed me how to derive the double angle identities from what I think is Euler's formula: e^{ix} = \cos x + i\sin x = e^{i2x} = \cos 2x + i\sin 2x = (e^{ix})^{2} = (\cos x + i\sin x)^{2} I have a question about this step...I...
  40. N

    Complex Analysis - Solving Complex Trig functions

    Homework Statement Now, I know there's two ways to go about this and it seems everywhere I look around on the web people are solving it in a way I think that seems longer, harder and more prone to mistakes in exams. It involves using the exponential identities and taking logs. I was shown...
  41. S

    On which quadrants are each of the six inverse trig functions defined?

    I have researched this area a little bit and now I am a little worried because three different websites have gave me three different answers. Some functions matched, but others didn't. My general consensus is inverse Sin= 1 and 4 quad inverse Cos= 1 and 2 quad inverse tan= 1 and 4 quad...
  42. M

    Inverse Trig Functions as a (unique?) solution to a PDE

    Hi, I know from basic math courses that inverse trig functions are multi valued (e.g. arctan(c)=θ+n*2∏). Now, if I solve a partial differential equation and I get an inverse trig function as part of my solution, does that mean solutions to the pde are non-unique? For example, if...
  43. B

    Solving an Odd Function with Periodicity

    Homework Statement Basically, I had a test yesterday and one of the questions was: "an odd function f(x) has a period τ=7. What is the value of f(75)-f(-30)" Homework Equations n/a The Attempt at a Solution I used periodicity to reduce = f(75-70) + f(30-28) = f(5) + f(2)...
  44. B

    Integration of inverse trig functions

    Homework Statement This is an integration of an inverse trig function. I don't see how they go from 1/2 to 1/4. I understand how they get the 1/2, du = 2dx, divide both sides by 2, but where does the 1/4 come from?
  45. B

    Simplifying multiple trig functions into a single trig function for physics II

    Homework Statement It has been a while since I have really been involved in trig seriously, But I felt it appropriate to go in this particular forum because in my classes from years back "precal" was the title associated with trig (: The Problem: sin(X) / sin(X/2)...
  46. D

    Limits of Trig Functions: How to Solve Using L'Hopital's Rule?

    Homework Statement lim(x -->0) (1-cos(14x))/(xsin(18x)) Homework Equations None? The Attempt at a Solution The hint tells me to use L'hopital's rule through which I got lim(x-->0) (sin(14x))/(18xcos(18x)+sin(18x)) (I factored out the 14 in the numerator) That gave me a 0/0 so I did...
  47. G

    How do you find the X-values of inequalites involving trig functions?

    Homework Statement What values of X between 0 and 2 pie radians satisfy each of the following: 1. |sinX|<0.5 2. |cosX|>0.5 Homework Equations The Attempt at a Solution Well the values of X lie between 1. -0.5 < sinX <0.5 2. cosX< -0.5 and cosX>0.5 How do you find the...
  48. J

    Derivative of inverse trig functions.

    Homework Statement Find the derivative: y=sec-1(1/2t3) Homework Equations \frac{\frac{du}{dx}}{|u|\sqrt{u^2-1}} The Attempt at a Solution I have an example to follow, but I don't know how step 1. became step 2.?...or more exactly the last part under the radical? (1-4t^6) instead...
  49. J

    Exact values of trig functions.

    So basically I know what the answer is to the problem and the steps on how to get there, but during the steps I'm not sure why one thing happens. sec(7pi/6) =1/(cos7pi/6) =1/(-cos(pi/6)/ <--- I'm not sure why the 7 disappears in this step and the cos becomes negative. I get the feeling...
  50. P

    Finding the period of 2 multiplied trig functions

    I am trying the period of 2 cosine functions that are multiplied with each other, but I am blanking out on how to find them. For example, given a function like: x(t) = cos(10*pi*t)cos(20*pi*t) I know it has something to do with the frequency of both functions (10*pi & 20*pi), but I...
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