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

    Improving Trigonometric Integration

    Homework Statement \int\frac{secx}{(tanx)^2}dx The Attempt at a Solution I tried all the u subs u=tanx and u=secx but neither worked. Should I used other methods? Please help me with the start!
  2. 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...
  3. T

    Can You Solve This Trigonometric Equation 2 Homework Problem?

    Homework Statement tan^2x = 2tanx*sinx Homework Equations N/A The Attempt at a Solution tan^2x = 2tanx*sinx (tan^2x)/tanx = (2tanx*sinx)/tanx tanx = 2sinx sinx/cosx = 2sinx cosx = 1/2 x = pi/3 \alpha = pi/3 and 5pi/3 Not sure what I'm doing wrong. The...
  4. T

    Solving Trigonometric Equations: Divide by Cosine

    Homework Statement sinx + cosx = 0Homework Equations N/AThe Attempt at a Solution sinx + cosx = 0 sinx = -cosx sinx = (+/-) sqrt(sin^2x - 1) (sinx)^2 = (+/-) sqrt(sin^2x - 1)^2 sinx = sin^2x - 1 Not really too sure what to do from here on. The answer is 3\pi/4 and 7\pi/4. I appreciate the...
  5. I

    Fairly basic trigonometric equation

    Acos(A*x)-Bcos(B*x)=0, where A>B Is there a general solution for an equation of this form? Thanks, intangible
  6. S

    Solving a trigonometric equation

    Homework Statement Determine the points of intersection for the two given functions on the interval 0<x<4pi y = 2 \sin \frac{x}{2} y = 3 \cos \frac{x}{3} 2. The attempt at a solution Well i tried graphing it and found out that the solution must lie somewhere between pi and 2pi...
  7. 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...
  8. quasar987

    Trigonometric Polynomial Form: How Does the +1 Factor Fit in T_n(x)?

    Homework Statement Maybe I'm just blind, but how is T_n(x)=\left(\frac{1+\cos(t-a)}{b}\right)^n of the form T_n(x)=\sum_{k=-n}^nc_ke^{ikx} ? I can get \left(\frac{\cos(t-a)}{b}\right)^n by setting c_{\pm n}=\left(\frac{e^{\mp ia}}{2b}\right)^n and the rest of the c_k= 0, but how does...
  9. S

    Can you help me with these tricky trigonometric identities for my math final?

    I need some help on 15 identities problems to help me study for my math final. They are blurry and somewhat hard to read, but if anyone wants to take a crack at some of them they are here: http://img407.imageshack.us/img407/5696/math1lh2.jpg...
  10. S

    Trigonometric Form and the Surprising Result: 1=-1?

    Hi, as I was studying complex numbers today I came across this, and I couldn't explain it: 1=e^{i0} 1=e^{i2\Pi} 1^{1/2}=e^{i2\Pi/2} 1=e^{i\Pi} 1=-1 Where is the mistake? Thank you very much for your help.
  11. W

    Roots of Trigonometric polynomials?

    I remember learning an iterative method that gives the answer to trigonometric polynomials such as sin(x)-0.7-0.611cosx = 0 where x is the angle in degrees. The person who I learned this method from called it the method for solving transcendentals. Now I can't seem to find any...
  12. T

    Range of a trigonometric function

    Homework Statement Find the range of the function: y=(2cosx+1)/(2cosx-1) algebraically Homework Equations Reducing it, I obtained: y= tan(3x/2)/tan(x/2), but the discontinuity confuses me The Attempt at a Solution I did it with my calculator and this is the result: Ran =...
  13. A

    Series with Trigonometric funtions

    [SOLVED] Series with Trigonometric funtions Homework Statement Determine whether the series converges conditionally,converges absolutely or diverges. \sum_{n=2}^{\infty}\frac{\sin (n+\frac{1}{n})}{\ln (\ln n)} The Attempt at a Solution \frac{\sin (n+\frac{1}{n})}{\ln (\ln...
  14. 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...
  15. G

    Trig Limits: Does x/sin(x) = 1 as x-->0?

    This isn't really a homework question--but just something I was thinking about while doing homework. Since lim(sinx/x) as x--->0 = 1 does that mean that lim(x/sinx) as x--->0 = 1 ?
  16. A

    Integration by Trigonometric Substitution.

    I'm not sure about answer.It looks very strange. Homework Statement \int_{1}^{e}\frac{dx}{x\sqrt{1+ln^2x}} The Attempt at a Solution for u=lnx-->u'=1/x \int \frac{du}{\sqrt{1+u^2}} substituting u=tan\theta =\int \frac{d\theta}{cos\theta}=ln|sec\theta+tan\theta|...
  17. 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...
  18. 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...
  19. tony873004

    Where does the x/3 come from in trigonometric substitution?

    This is an example from the book. Evaluate \int {\frac{{\sqrt {9 - x^2 } }}{{x^2 }}dx} I understand all the steps that get me up to = - \cos \theta \, - \theta \, + C Then the book goes on to explain: "Since this is an indefinate integral, we must return to the original variable...
  20. 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...
  21. L

    Integrating \sqrt{16-(2x)^{4}}xdx using Trigonometric Substitution

    \int\sqrt{16-(2x)^{4}}xdx Hint says you may like to use the identity sin(theta)cos(theta)= sin(2theta)/2 However, I think I found a way to use 1-sin^2(theta)=cos^2(theta) First, (2x)^4 = 16x^4 So make it 16(1-x^2)^2. Take the 16 out of the root and the integral and you have...
  22. S

    Proving trigonometric equation?

    for x between 0 and 2pie, solve cos(4x)=sin(2x)... is this a proving trigonometric equation? i don't think it is
  23. J

    Trigonometric sequence - Find Un out of Un+1 - hard

    Homework Statement Un a sequence 0 < x < pi/2 U0 = 2cosx Un+1 = quroot( 2 + Un) 1) Calculate U1, U2, and U3 in function of x (simplify maximum) 2) Show that Un = 2cos( x / 2^n)Homework Equations The Attempt at a Solution 1) U1 = sqroot( 2 + 2cosx) = sqroot( 2 (1 + cosx)) = ?? (now , how...
  24. C

    Solving Trigonometric Equations - 0 to 2pi

    Question 1 sin theta = 2/3 tan theta < 0 sin2 theta = ?? how would i do this? please help ____ Question 2 solve 2cos^2 theta - cos theta = 1 for 0 <= theta < 2pie how would i do this one too?
  25. H

    Trigonometric Deduction from Euler's Formula: Finding the Correct Relations

    Anybody that can show how to deduce cos(a+b) = cos(a)cos(b) - sin(a)sin(b) sin(a+b) = sin(a)cos(b) + sin(b)cos(a) From the relations that we get from eulers formula.. Should be really simple but I think that I have got some relations wrong so I need to se the real solution.:uhh: Thanks!
  26. 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...
  27. H

    Deriving a trigonometric identity

    For a homework assignment I'm supposed to prove that sin(x)^2+cos(x)^2=1, using only the following identities (along with algebraic operations): sin(-x)=-sin(x) cos(-x)=cos(x) cos(x+y)=cos(x)cos(y)-sin(x)sin(y) sin(x+y)=sin(x)cos(y)+cos(x)sin(y) I can't figure this out, because as far...
  28. S

    Solving an Equation with Trigonometric Terms

    more trig... Amm I've been at this problem for an hour already it loooked really easy but for some reason i can't reach the answer (tan x + sec x)^2 = (1 + sin x)/(1 - sin x) soo what went and tried was expand it and then exchage tan and sec.. (sin^2 x /cos^2 x )+ (2/cos x) + (1/ sin^2 x)...
  29. P

    How to solve the trigonometric equation sin(x) = x/2?

    How would one go about solving sin(x) = x/2 I.e. the intersections of f(x)=sin(x) & g(x)=x/2 I can rigorously solve this by going to each individual period and finding the intersections. But is there a better way?
  30. K

    Transforming Trigonometric Identities: Solving 2(cosx)^2+2cosxsinx=0

    Which trigonometric identitiy are used to transform 2(cos(x))^2+2cos(x)sin(x) into 0?
  31. G

    Trigonometric identity forced oscillations

    Homework Statement http://www.jyu.fi/kastdk/olympiads/2004/Theoretical%20Question%203.pdf http://www.jyu.fi/kastdk/olympiads/2004/Solution%203.pdf Question A- (b) They use some trigomentric identity that I don't understand, which one is it? Thanks in advance. Homework...
  32. J

    How to Prove Trigonometric Reduction Using Integration by Parts

    Homework Statement Prove the following trigonometric reduction using integration by parts: \int \sin^n x dx = - \frac{\ \sin^{n-1} x \cos x}{n} + \frac{\ n-1}{n} \int \sin^{n-2} x dx 2. The attempt at a solution I tried using integration by parts by breaking up sin^n x into sin^(n-2) x...
  33. T

    Trigonometric graph question

    just encountered this question and kinda confused at how to solve it since I havn't bin told and havn't worked it out for myself. hope you can help. Homework Statement Use the graphs (shows 2 graphs) to find the values of x in the range 0 /leq x /leq 720 when 2sinx = cosx -1 Homework...
  34. dextercioby

    What is the integral of tangent x divided by x from 0 to infinity?

    Homework Statement Compute \int_{0}^{\infty} \frac{\tan x}{x} {} dx Homework Equations The Attempt at a Solution Got no idea, obviously the derivation differs from the \int_{-\infty}^{\infty} \frac{\sin x}{x} {} dx one. The result I'm supposed to get is \frac{\pi}{2} .
  35. P

    Solving Trigonometric Equations

    Homework Statement use the inverse functions where necessary to find all solution of the equation in the interval [0,2pi). Use a graphing utility to verify. 2cos(squared)x+3cosx=0 Homework Equations The Attempt at a Solution The answer is pi/2 and 3pi/2 from my teacher.. ===...
  36. N

    Vectors and the trigonometric function

    Hey I'm just having trouble with this question Two vectors have magnitudes of 10 and 15. The angle between them when they are drawn with their tails at the same point is 65 deg. The component of the longer vector along the line perpendicular to the shorter vector, in the plane of the vectors...
  37. R

    Deriving Trigonometric Functions: How to Find the Derivative of tan4x

    How do I get the derivative of y=tan4x The answer in the back of the textbook says (4/cos^2(4X))
  38. 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...
  39. A

    Trigonometric Integral excersice

    Homework Statement Integral of \int \sin^{11/3}\alpha\, d\alpha Homework Equations \sin^2\alpha = 1 - cos^2\alpha The Attempt at a Solution \int (\sin^2\alpha)^{4/3}\sin\alpha \, d\alpha \int (1-cos^2\alpha)^{4/3}\sin\alpha \, d\alpha u = \cos\alphad du = \sin\alpha\...
  40. C

    Proving this trigonometric identity

    Homework Statement Prove the following identity: (1 + cos \theta)^2 + sin^2\theta = 2(1 + cos \theta) Homework Equations cos^2 \theta + sin^2 \theta = 1 The Attempt at a Solution I've squared out the first bracket so that it becomes 1 + cos^2 \theta and multiplied out the second bracket so...
  41. I

    Simplifying Trigonometric Problems | Proven Equations & Solutions

    Homework Statement Simplify The problem is either 1+[(cosx)/2] or 1+[cos (x/2)] The first one looks unworkable so I'm going with the second...unless any of you see that the first one looks normal... Homework Equations I derived/proved some below... The Attempt at a...
  42. J

    PLease HELP with this trigonometric equation

    Greetings my friends: I have been reading a book about optimization and I found the following trigonometric equation: tan(10x.pi)= - 10 pi x (this equation goes from x E [-1,2] it is easy to see that has infinite solutions, but the author came to the conclusion that the solutions are...
  43. D

    Proving a Trigonometric Identity Using Exponential Functions

    Hi. I need to prove the following identity \arccos{z} =i \ln { z + (z^2 -1)^\frac{1}{2} } I was given a hint to write \cos{A}=z, then rewrite \cos{A} in terms of the exponential. \cos{A}=\frac{\exp{iA}+\exp{-iA}}{2}=z I took the log on both sides and got stuck at that...
  44. K

    Trigonometric Identity in My Book: Understanding (cos4x)^2 = 1+cos8x

    In my book, (cos4x)^2 is written 1+cos8x without referring to any formula. Which trig. identity is used here?
  45. K

    Trigonometric Identity Problems

    Can anyone help me solve the following problems? sec theta -1/1-cos theta = sec theta tan (pie/2 - theta) tan theta =1 Thanks
  46. C

    Complicated trigonometric equation

    I need to solve sin(ax)sin(bx) - k cos(ax)cos(bx) = -1 where k > 1 is constant and so are a and b, and they are all irrational. (It's part of a much larger question in optics...) Any ideas, please? Thanks
  47. P

    Solving simultaneous trigonometric equations

    Hi, all I am new to this site and I was wonder if anyone could help. I took linear algear algebra last semster and am currently taking statics. I want to know if I can use some of the techniques I learned in linear algebra to solve simtultaneous equations which involve sine or cosines For...
  48. 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...
  49. 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...
  50. S

    Factoring trigonometric equation

    Hi, I'm having trouble understanding how my math book factored the following equation. [(cos(t)sin(t) - (1 + sin(t))cos(t)) / (cos(t)cos(t) - (1 + sin(t))sin(t))] to get [(cos(t)(1+2sin(t))) / ((1 + sin(t)) (1 - 2sin(t)))] I get the numerator but I don't understand what happened...
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