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. T

    Finding the Derivative of Trigonometric Functions with Exponents?

    hey I have this question and have looked it up in the textbook and web sites but can't seem to find what to do! Any assistance would be appreciated thanks! Find the first derivative w.r.t the relevant variable 5^(sin(theta)) I am guessin the relevant variable is theta but I don't even no...
  2. 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}
  3. S

    Looking for Trig Identities Review for Calculus? Check These Out!

    Hi, I have an exam on trig integrals tomorrow and need to freshen up on some basic trig rules (i.e. d/dx of sin(mx) and trig identities such as sin^2(x)+cos^2(x)=1) I was curious if anyone new of some good websites that reviewed all common trig identities used in calculus. Any help is...
  4. L

    Derivatives of Trigonometric Equations

    Alright, moving on to the topic of taking the derivative of a trigonometric equations, I have been given the problem to find the equation of the tangent line to the curve at the given point for: y = 1/sinx+cosx at (0,1) Now we know that the equation is y-1=m(x-0), so I've tried solving for...
  5. V

    Trigonometric Integral Help: Solving for a<1 Using Complex Contour Method

    need some assistance with the following integral: \int_0^{2\pi} cosx/(a-cosx), a-parameter (say a>0) i've converted it into a complex contour integral over z=e^(ix): ~ \int_{|z|=1} dz (z^2+1)/[z(z^2-2az+1)] which is easily evaluated for a>1. my question regards a<1 - i am not sure...
  6. N

    Inverse trigonometric function

    Find dy dx 1.) y = In __x2 (x+1)___ (x + 2)3 2.) y = x3 ( 3lnx-1) 3.) y = __cos6 2x__ (1-sin2x)3 4.) y = __tan 2x__ 1- cot 2x 5.) y = x e (exponent pa po ng e) sin2x 6.) Arc tan...
  7. C

    Trigonometric Identity for 1/2csc(THETA)sec(THETA)

    Hi, I have a question about a problem: 1/2csc(THETA)sec(THETA) I have to find the identity for it and I know that csc is 1/sin and sec is 1/cos but after that I don't see where it can lead to a simple answer. If anyone knows it would be helpful, thanks!
  8. K

    Are c1 cos(wt) + c2 sin(wt) and A sin(wt + phi) Equivalent?

    Hi. I have to show that x(t)=c1 cos(wt) + c2 sin(wt) '(1)' and x(t) = A sin(wt + phi) are equivalent. I know I have to use sin(alpha + beta) = sin(alpha)cos(beta) + cos(alpha)sin(beta) or cos(alpha +beta)= cos(alpha)cos(beta) - sin(alpha)sin(beta) I have been strugling with this...
  9. K

    Trigonometric Identities- simplify sin^4x - cos^4x

    hey just wonderin if any1 could give me a hint as to the best method to prove the following trigonometric indentity: sin^4x-cos^4x = 1 - 2cos^2x i tried the side more complicated first...but can't seem to hav any luck...other then maing it more complicated! umm the x's are meant to be...
  10. 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...
  11. K

    Trigonometric Functionssimplify sin squared functions

    just looking at another question to do with trigonometric functions and I can't see how they simplify the follwing: 2sin^2x-3sinx-2=0 to (2sinx+1)(sinx-2)=0 again i prob thinking sumthin really stupid...but i can't see wat! cheers
  12. O

    Derivative of inverse hyperbolic trigonometric functions

    I'm working on a pre-freshman year math packet for college, and at one point it asks for the derivative of sinh-1(x), followed up by the derivative of ln( x + sqrt(1+x2) ). In high school, we never really covered hyperbolic trigonometry, but I have previously derived that the inverse of sinh is...
  13. E

    Can Trigonometric Series be Evaluated Using Euler-Maclaurin Sum Formula?

    Let be the series: \sum_{n} e^{if(n)} where f is a function perhaps a Polynomial ..then my question is..how can this series to be evaluated (at least approximately) ?..perhaps using Euler-Bernoulli sum formula, and another question what are they used for?, i heard in a book that Goldbach...
  14. T

    Solving trigonometric equations

    as we know that (sin(x))^2 + (cos(x))^2=1 how about (sin(3x))^2 + (cos(3x))^2 and 5sin(3x)^2 + 6cos(3x)^2?? how can we solve these problems?? thanx
  15. E

    How Can I Graph THis trigonometric Function?

    Cud u Help Me How to Graph these? y = 2 + sinx ; where x = teta y = 2sinx ; where x = teta cud u give me some ideas pls?
  16. Orion1

    Iterated trigonometric differentiation

    These are some equations that I recently developed and submitting for review. Evaluations?, comments? Iterated trigonometric differentiation: \frac{d^n}{dx^n} \sin x = \sin \left(x + \frac{n \pi}{2} \right) \frac{d^n}{dx^n} \cos x = \cos \left(x + \frac{n \pi}{2} \right) Iterated...
  17. S

    Differentiation of Inverse Trigonometric Functions

    Hey guys, I have to know how to Differentiate Inverse Trigonometric Functions in my next exam and need somewhere to study up on them. Do you know of any web sites I could read? Can't find anything on Karl's Calculus. Thanks
  18. S

    Solve Trigonometric Equations

    Dear all, How to solve this trigonometric equation systematically? 2cos^(2) x - cos x = 0, where x ∈ [0,360] Thanks in advance
  19. T

    Trigonometric functions like sin(2x)=2sin(x)cos(x)

    Hi guyz, as we know we have some known relations in the trigonometric functions like sin(2x)=2sin(x)cos(x) and sin(x/2)=1/2-1/2 cos2x My question is are there similar formulas for arcsin and arccos? I know those only ! arcsin x =ln(ix-sqrt(1-x^2)) arccos x =ln(-ix-sqrt(1-x^2))...
  20. F

    Applying trigonometric functions to some real life situations

    I'm having some trouble with applying trigonometric functions to some real life situations, particularly this one problem in my homework. Andrea, a local gymnast, is doing timed bounces on a trampoline. The trampoline mat is 1 meter above ground level. When she bounces up, her feet reach a...
  21. P

    Trouble with Trigonometric Integral? Get Help Now!

    Haven't done integrals in such a long time and now I'm having some trouble with this question here. Any help would be appreciated. Thanks :smile: http://img331.imageshack.us/img331/4333/screen192cj.jpg
  22. P

    Solving a Trigonometric Equation: Tips and Tricks for Beginners

    I have stuck on this problem for long time sin^2 \alpha = \frac{\alpha}{2} I never meet this kind of problem before, and I have no idea about this. Could someone tell me how to solve this kind of problem? Thanks in advance. (Ans: \alpha = 1.39 rad )
  23. J

    Trigonometric Identity? - Deriving the Double Angle Formulas for Sine and Cosine

    Trigonometric Identity?? I just can't figure this out. I don't think I've covered enough material to do this. Can anyone help? I've put the entire question but I am sure all i need is a little explanation and maybe the first answer and i could do the rest. :confused: Let z =...
  24. P

    Can this trigonometric equation be solved for x_1 and x_2 in terms of \alpha?

    x_1(cos\alpha-1) + x_2sin\alpha = 0 x_1sin\alpha + x_2(-cos\alpha-1) = 0 How to solve this equation? Can anyone help me?
  25. Loren Booda

    Polynomial, trigonometric, exponential and fractal curves

    What other curves are there that cannot be described by the above? Are trigonometric functions actually a special case of exponentials with complex powers?
  26. H

    Verifying a Hard Trigonometric Identitiy ()

    My Pre-Cal teacher gave us this problem today. I have worked on it for a very long time and have goten no where :confused:. I was wondering if anyone had any ideas on how to do it, or even where to start. I started it myself with using the pythagorean idenities for the left side, then the...
  27. C

    Help with Trigonometric Simplifying

    Hi, i need help with simplifying this problem. I think it is an identity, but it looks very complex, and I wanted some other peoples thoughts/opinions. Well here goes... cos^2((pi/4)-(x-2)) - sin^2((pi/4)-(x-2)) I reduced it using the identity cos2x=cos^2x - sin^2x. I came out with...
  28. O

    Solving Trigonometric Equations

    I have a question when solving trigonometric equations. For example: Find all the solutions in the interval [0,2pi) \sin \theta \tan \theta = \sin \theta \] If you choose to divide through by \sin \theta\] we get, \tan \theta = 1\] such that \sin \theta \ne 0\] otherwise we are...
  29. S

    Struggling with Trigonometric Equations: Need Some Help?

    I'm having trouble with two problems: 2tan(x) - 2cot(x) = -3 and cos(x)^2 + sin(x) = 0 On the 2nd one, I can substitute 1-sin(x)^2 for cos(x)^2 right? I tried that, but it didn't work. And I have no clue what to do on the first one. Little help please?
  30. H

    Derive most trigonometric identities from the addition formulas

    In the same way that it is possible to derive most trigonometric identities from the addition formulas, what is the way that the difference of sines and cosines formulas were derived, such as \sin{a}-\sin{b}=2\cos{\frac{a+b}{2}}\sin{\frac{a-b}{2}} thanks, I am trying to avoid as much...
  31. S

    Can substitution be used to solve these trigonometric integrals?

    the two problems are the integral of (sec^3x)(tan x)dx and the integral of (sec^4x)(tan x)dx will substitution work for both of these problems
  32. A

    Derive a trigonometric equation for the volume of the cone

    A circular cone is inscribed in a sphere with a radius of 30cm. The semi vertical angle is theta. Derive a trigonometric equation for the volume of the cone. This has be stumped. I tried looking up proofs for the expression of the volume of a cone for inspiration but all involve calculus.
  33. D

    Limits of a Trigonometric Function

    Question: lim(x->0) for (tanx - sinx) / (sinx)^2 This is what I got: = (sinx-sinxcosx) / (cosx)(sinx)^2 = (sinx)(1-cosx) / (sinx)(sinx)(cosx) = (1 - cosx) / (sinx)(cosx) However, I can't figure out what to do from this step, as the limit still equals 0/0 at this stage.
  34. D

    Limit of a Trigonometric Function

    Question: lim(x->0) for (tanx - sinx) / (sinx)^2 This is what I got: = (sinx-sinxcosx) / (cosx)(sinx)^2 = (sinx)(1-cosx) / (sinx)(sinx)(cosx) = (1 - cosx) / (sinx)(cosx) However, I can't figure out what to do from this step, as the limit still equals 0/0 at this stage.
  35. S

    How Do You Integrate sin^6(x) Using Trigonometric Identities?

    Problem: \int sin^6 x dx Progress so far: \int (sin^2 x)^3 dx \frac{1}{8} \int (1-cos2x)^3 dx \frac 1 8 \int (1 - 3cos2x + 3cos^22x - cos^32x) dx Any help is appreciated. I can see using a half angle identity for cos^2(2x), but what do I do with the cos^3(2x)? Steve
  36. C

    What is the purpose of trigonometric identities

    As I rack my brains to solve(or atleast pretending) to prove Trig identites. What is it application in real life? :cry:
  37. A

    Understanding Periods of Trigonometric Functions with Different Frequencies

    f(x)= sin 3x - (1/2)sin x, find the period. i know the period for sin 3x is 2pi/3 and the period of sin x is 2pi but how do you subtract these? I totally forget how to do this! I mean i could find the answer with any graphing program but i want to know how to do this type of problem.
  38. C

    Solving a Trigonometric Equation: Primary to Secondary Solutions

    How do I get from the primary to the secondary solution of a trigonometric equation? This book tells me that the second angle is within -\pi \leq \theta \leq \pi, in a different quadrent, but I don't follow :\ Thanks. Edit: I got it (I think!): I can pick the correct quadrent using the...
  39. T

    Turning Equations into Trigonometric Equations

    Hello all, I have been looking up the golden ratio and found most of what I needed on mathworld. The site states that \phi \ = \ \frac{1}{2}(1+\sqrt{5}). I can see how (despite the fact that I don't understand how the ratio: \phi \ = \ \frac{AC}{BC} \ = \ \frac{AB}{AC} is formed but...
  40. H

    Trigonometric differentiation

    When y=sin(pi*x), why does y'=cos(pi*x)*pi, not y'=cos(pi*x)? hk
  41. J

    Simultaneous trigonometric equations

    I am looking for help in solving a pair of simultaneous equations. I have not come across any maths book that solves trigonometric ones. I was wondering if I could get a step by step solution. Thanking you in advance for your time: 5.4=10cos(x) + 13.41cos(y) ....(i) 0=10sin(x) +...
  42. Pyrrhus

    Calculating Angle C in Vertical Plane: Trigonometric Problem in Dynamics Book

    Hello, I'm in need of a hint or few pointers on how to calculate the angle C of the picture attached. I've already calculated y. I was doing a few problems in this Dynamics book, i bought recently, and the ascention angle (angle C) is beating me :eek: "The airplane C is being tracked...
  43. L

    Pre-Test on Trigonometric Equations and Applications

    I have a math test on the chapter on Tuesday, and my teacher handed out the pre-test on Thursday. There are a few problems I am totally stumped on, and figured the math geniuses here could give me some help. Some I can get somewhere with, some I don't know where to begin. Here is one...
  44. N

    Simple Integral w/ Trigonometric Substitution

    Hello everyone, I am having some trouble with an integral. \int \sqrt{x^2 - 1} dx so far: x = sec \theta \frac{dx}{d \theta} = sec \theta tan \theta dx = sec \theta tan \theta d\theta now we substitute: \int \sqrt{x^2 - 1} dx = \int \sqrt{sec^2 \theta - 1} sec \theta tan...
  45. C

    What is the correct answer for this trigonometric improper integral?

    Here's a integral where I have to use trigonometric substitution but I can't get the right answer. [int a=0 b=3] 1/(sqrt[9-x^2]) dx I did the limit as t approches 3 from the left. Then i did my trigonometric substitution, and it gives me arcsin(x/3). Then i computed what i had...
  46. N

    Solving a Trigonometric Equation

    How would you solve for \alpha in the following equation? 4=\cos(\alpha)+\cos^2(\alpha)+\cos^4(\alpha)
  47. L

    Chapter Summary: Trigonometric Functions

    The problem reads: Find \sin\theta and \cos\theta Part a gives me the coordinates \left(-1,1\right) The triangle I got had the x-length as -1, while the y-length was 1. The hypotenuse I got was \sqrt{2} Since \sin is \frac{opposite}{hypotenuse} I got \sin\theta=\frac{1}{\sqrt{2}}...
  48. A

    Trigonometric Substitution

    \int x^3\sqrt{4-9x^2}dx I tried to use x=\frac{2}{3}\cos{(x)} but it just left me with \int \sin^3{(x)}\cos^2{(x)}dx Any suggestions? Thanks for your help.
  49. L

    Inverse Trigonometric Function Problems

    The math book I have does a pretty terrible job explaining this to me, because I am absolutely stumped as to why I get every question wrong in two sections: finding values of each expression in radians (can often be given in terms of ?) and finding approximate/exact values of the expressions...
  50. D

    Is there an easier approach for solving trigonometric identities?

    I'm having problems with it at school lately, I am not going to layout every single problem and ask for help. I am just wondering if there is a better approach to it rather than trying to solve one side in order to get it to equal the other side. For example, sin^2 x + cos^2 x = -cos^2 x - sin^2...
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