What is Trigonometric: Definition and 1000 Discussions

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.Trigonometry is known for its many identities. These
trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.

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  1. A

    How Do You Solve Trigonometric Equations Involving Cosine and Sine?

    Hi All, I’m currently attempting to work ahead on my precalculs and I’m looking at trigonometric equations. I seem to have a bit of a problem with this example (i haven’t had too many issues with the rest of the exercises), 9cos(2x)+sin(x)=9 (solve for x in the interval 0 <=x <=2pi.) I’m...
  2. 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)...
  3. D

    Finding equation for trigonometric function based on two points

    Is there any standard way to do this? Specifically, I know the equation I'm looking for is a cosine function, and I know two points. Is there a way to find the equation of the cos function which has been translated and compressed given only these two points?
  4. 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...
  5. N

    How do you simplify the integral after trigonometric substitution?

    I’ve got this tricky trig substitution that I’ve been trying to do as for my Calc II class. I think that I’ve done the substitution part right I just have no idea where to go from here. I’ve tried many routes including integrating by parts as well. Please help me thank you...
  6. 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...
  7. A

    How Can I Solve This Trigonometry Problem Correctly?

    Homework Statement http://img135.imageshack.us/img135/4150/trighelp.jpg Homework Equations The Attempt at a Solution http://img18.imageshack.us/img18/8480/workz.jpg Please help I'm totally confused, if anyone can help I can offer a reward of monetary value :).
  8. A

    How to Solve Trigonometric Word Problems Correctly?

    Homework Statement http://img135.imageshack.us/img135/4150/trighelp.jpg Homework Equations The Attempt at a Solution http://img18.imageshack.us/img18/8480/workz.jpg Please help I'm totally confused, if anyone can help I can offer a reward of monetary value :).
  9. M

    Finding points on a trigonometric function

    Homework Statement Use the function f(x) = sin x + cos x, where 0<=x<=pi to answer the following. a) Determine all stationary points. Classify the points as minimum or maximum b) locate the points of inflection c) determine the endpoints of the interval Homework Equations...
  10. 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...
  11. 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...
  12. P

    How do I solve for the solutions of sin 2θ = sin θ in degrees?

    Homework Statement sin 2θ = sin θ Find the solutions in degrees. Homework Equations sin 2θ = 2sinθcosθ The Attempt at a Solution sin 2θ = sin θ 2sinθcosθ = sin θ sinθ *cosθ/sin θ = 1 That's as far as I can get, and I think that is wrong. How do I procede from 2sinθcosθ = sin θ?
  13. I

    Trigonometric Equation with Multiple Functions and Arguments

    Can anyone help solve the below for 'a' 0=b(ycos(a)-bcos(2a)+xsin(a)) I've reduced it to b=ycos(a)+2sin(a)sin(a)+xsin(a) =sin(a)(cot(a)+2sin(a)+x) Which I think is right, but I can't get any further without just ending up with different forms that appear more complex.
  14. 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...
  15. K

    Trigonometric Substitution Problem - Calculus 2

    Hey there This is a trig substitution for my Calculus 2 class and I really have NO idea how to get started... \int\frac{4}{\sqrt{3-2x^2}}dx My professor has yet to go over how to evaluate trigonometric substitutions with coefficients in front of variables.
  16. N

    Solving trigonometric equation

    Hi guys, I need to solve the following trigonometric equation. Some time ago I would have used Fortran with the appropriate libraries but now I was wondering whether there is any better approach. Any suggestion? w^2/c^2= k//^2 - Sqrt(w^2 Eps_b^2/c^2 - k//^2) * Tan[Sqrt(w^2 Eps_b^2/c^2 -...
  17. N

    A Generalized trigonometric identity for Cos(x_1++x_n)?

    Hi Everyone, Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again? If it exists then what is about Sin(x_1+x_2+...+x_n)? [It is understood that there will be 2^(n-1) number of...
  18. M

    Solving a trigonometric equation help

    Homework Statement sqrt3 cot 3x + 1 = 0 where 0 _<x_<2pie Homework Equations The Attempt at a Solution I narrowed it down to a constant on one side, cot 3x = -1/sqrt3 which I could turn into cos x/sin x = -1/sqrt3 but I have no idea where to go from here. Any help...
  19. M

    Proving Trigonometric Identites

    Homework Statement tan x + 1/tan x - 1 = sec x + csc x/sec x - csc x Homework Equations The Attempt at a Solution Ive tried working with the right side and have gotten as far as sin x/cosxsinx - cos x/cosxsinx Im not sure if I am just out to lunch, or on the right track...
  20. F

    Proving (tan2x) - (sin2x) = (tan2x)(sin2x): Trigonometric Proof Help

    Homework Statement (tan2x) - (sin2x) = (tan2x)(sin2x) Prove these two are equal. Homework Equations Various trig identities, mainly tan2x = sin2x / cos2x The Attempt at a Solution I tried putting in Sin2x / cos2x for tan, but I don't know where to go from there. Anyone mind...
  21. M

    How can I solve the Trigonometric First Order DE \frac{dx}{dt} = \cos(x+t)?

    \frac{dx}{dt} = \cos(x+t) I'm having real troubles with this; I tried a substitution of u=x+t but it just ends up as, t=\tan{u} + \csc{u} + C And I can't see where to go from there. (the middle function is cosec; it didn't come out very clearly on my screen).
  22. M

    Proving trigonometric identities using sum and difference formulas

    Homework Statement Prove that sin (a+b) does not equal sin a + sin b. Let a = pi/3 and b = pi/6 Homework Equations The Attempt at a Solution Where I am lost is how to figure out what pi/3 or pi/6 is. Like for example, how can I know sin pi/3 is sqrt3 over 2? Is this a...
  23. 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...
  24. 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...
  25. R

    Integral of inverse trigonometric function

    Homework Statement \int1/sqrt(1-(x+1)^2) dx 2. The attempt at a solution I think a=1 but don't know what to set u equal to.
  26. C

    Unsolvable limit of trigonometric function

    I need to find the limit of sin(3t)^2/t^2 as t approaches 0. We have not yet learned L'Hopital's rule, so how do I find the limit here? I tried to take the derivative of sin(3t)^2/t^2, but it is nowhere near cancelling out 't' from the bottom. Thanks
  27. T

    Express Trigonometric Identities of Terms of Variables

    Homework Statement Be x an element in the interval [Pi/4, 3Pi/4] express cos(2x), sin x, sin (x+pi) in terms of x. You must know that, for this question, cos x = z and z will always be < 0. Homework Equations cos(2x) = 2 (cos(x))^2 - 1 cos(2x) = cos^2 x - sin ^2 x sin^2 x + cos ^2 x = 1...
  28. L

    Understanding the Trigonometric Identity: cos^2 x = 1/2 + cos(2x)/2

    Why is cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2} ?
  29. C

    Use trigonometric substitution to verify

    Homework Statement Using trigonometric substitution, verify that \int \sqrt{a^2[-t^2}dt (INTEGRAL FROM 0 TO \pi)=(1/2)a^2sin^{-1}(x/a)+(1/2)x\sqrt{a^2-x^2 Sorry it doesn't seem to want to allow me to place superscript inside a square root. but inside the first sq. root is a2-t2 and the...
  30. G

    Is the Trigonometric Homework Statement Solvable?

    Homework Statement prove : 1-cos x / sin x = sin x / 1+cosx 1-cosx / 1+cos x = tan^2 (x/2) Homework Equations The Attempt at a Solution I have No idea how to solve it
  31. C

    A trigonometric substitution problem

    Homework Statement \intsin3cos2xdx The Attempt at a Solution I've successfully solved this problem by factoring out 1 sinx and changing the sin2x to (1-cos2x then assigning u=cosx and du=-sinx and so on. What I'm wondering is why does letting u=sin3x in the original integral not...
  32. O

    What Are the Limits of Cosine and Tangent as Theta Approaches Zero?

    Hi everyone, I would like to know, since the lim theta --> 0 (sine theta / theta) = 1, what is the limit theta --> 0 of cosine and tangent? Thank you very much in advance
  33. 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...
  34. A

    Solving a trigonometric equation

    I'm trying to solve: sin2x = -cos2x so my first step was... 2sinxcosx = -cos^2x - sin^2x and then... 2sinxcosx + cos^2x + sin2x = 0 but I'm confused on how I would end up factoring 2sinxcosx + cos^2x + sin2x, or if I did something wrong earlier in the problem...thank you
  35. L

    Are the given trigonometric identities correct?

    Homework Statement tanX+tan Y / 1-tanXtanY= cotX+cotY/cotXcotY-1 Homework Equations tan= sin/cos cot= cos/sin The Attempt at a Solution
  36. L

    Can This Trigonometric Identity Be Verified?

    Homework Statement verify the identity : 1+sec(-∅)/sin(-∅+tan(-∅) = -csc ∅ Homework Equations The Attempt at a Solution 1+sec∅/-sin-tan∅ = -csc∅ I don't know where to start from, does anyone have any idea?
  37. H

    A confusing trigonometric question

    A confusing trigonometric question! Hi guys, I had a question. I was given the following equality. There was no problem statement that came with it (i.e. it did not say solve the equation or prove the identity, or anything of this nature) sec x + csc x / tan x + cot x = sin x + cos x The...
  38. 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...
  39. B

    Proving Trigonometric Identity: cosx/1-sinx + 1-sinx/cosx = 2secx + 2tanx

    cosx/1-sinx + 1-sinx/cosx = 2secx + 2tanx I can get to 2secx + tanx - tanx, any help is appreciated
  40. S

    Ferris Wheel - Trigonometric Function

    Pendulum Ride - Trigonometric Function Homework Statement At Canada's Wonderland, a thrill seeker can ride the Xtreme Skyflyer. This is essentially a large pendulum of which the rider is the bob. The height of the rider is given for various times: Time(s) 0 1 2 3 4 5 6...
  41. S

    Trigonometric identities for integral problem

    Homework Statement I have this integral to solve: \int \frac{ab}{a^2 cos^2 t + b^2 sin^2 t} dt The limits are 0 to 2*pi. Homework Equations The Attempt at a Solution I've tried using trigonometric identities, trigonometric substitution... and many kinds of algebraic...
  42. H

    How can trigonometric substitution be used to simplify a complex integral?

    Hello, I have been wokring on this problem for some hours now, and I get the wrong answer, but I can't understand why, could you guys please look at it? http://img219.imageshack.us/my.php?image=utennavnwv7.jpg
  43. C

    A Trigonometric Identity Probelm

    [SOLVED] A Trigonometric Identity Probelm If I have sin^2 2x would I be able to apply the identity sin^2x = (1/2)(1-cos2x) to get this: sin^2 2x = 2(1/2)(1 - cos^2 x) Similarly, if I had sin^2 2x + cos^2 2x would I be able to use the identity sin^2 x + cos^2 x = 1 to get: sin^2 2x +...
  44. S

    Application of Cauchy's formula for trigonometric integrals.

    Homework Statement ∫dθ/(1 + (sinθ)^2 ), [0, π] Homework Equations Cauchy's Formula, perhaps Cauchy's Thm. http://mathworld.wolfram.com/CauchyIntegralFormula.html http://mathworld.wolfram.com/CauchyIntegralTheorem.html The Attempt at a Solution I first substituted sinθ =...
  45. C

    Integration by trigonometric substitution

    Homework Statement Solve the integral using trigonometric substitution. \int\frac{\sqrt{4x^{2}+9}}{x^{4}}dx Homework Equations 2x=3tan\theta x=2/3 tan\theta dx=2/3 sec2\thetad\theta \sqrt{4x^{2}+9}=3sec\theta The Attempt at a Solution \frac{8}{9} \int...
  46. S

    What is the solution to this trigonometric limit question?

    Homework Statement Lim (1-cosX)/X^2 X->0 Homework Equations Not 100% sure try to use the following identity: sin^2(X)=(1-cosX)(1+cosX) or sin^2(X)=1-cos^2(X) The Attempt at a Solution Tried substituting (sin^2(X))/(1+cosX)) for 1-cosX but that didnt help and so for the last half...
  47. 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...
  48. P

    Trigonometric Substitution

    Homework Statement \int \frac{e^{3x}dx}{\sqrt{1-e^{2x}}} Homework Equations The Attempt at a Solution Alright so I am able to do other similar problems fine, I think it is the "e" that is throwing me off as well as the fact that the "x" is in the exponent. I started the...
  49. 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...
  50. 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...
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