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. G

    Calculus II - Trigonometric Substitutions - Evaluate integral dx/sqrt(1-2x^2)

    Homework Statement Evaluate integral dx/sqrt(1-2x^2) Homework Equations The Attempt at a Solution My Answer: -cos^(-1)(sqrt(2)x)/sqrt(2) + c See my work in attachment Wolfram Alpha Answer: (sin^(-1)(sqrt(2) x))/sqrt(2)+constant...
  2. G

    Calculus II - Trigonometric Substitutions

    Homework Statement Evaluate integral dx/sqrt(x^2-49) Homework Equations The Attempt at a Solution SEE POST #4 BELOW WITH ATTACHMENT TO VIEW MY SOLUTION EASIER TO READ i was trying to use a csc(theta) substitution and I don't see how my logic and math below is flawed but apparently I'm...
  3. G

    Calculus II - Trigonometric Substitutions, Explanation Please

    Evaluate integral sqrt(9-x^2)/x dx I get to here -3 ln|csc(sin^-1(x/2)+cot(sin^-1(x/3))|+3cos(sin^-1(x/3)+c I plugged in for theta because originally I had this -3 ln|csc(theta)+cot(theta)|+3cos(theta)+c and I set x equal to a expression x=3sin(theta) Hence i had to...
  4. G

    Calculus II - Trigonometric Integrals

    Homework Statement Evaluate integral csc(x)^4/cot(x)^2 dx Homework Equations The Attempt at a Solution Apparently I'm doing something wrong, what I'm not sure, thanks for any help My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c integral csc(x)^4/cot(x)^2 dx used fact that...
  5. G

    Calculus II - Trigonometric Integrals HARD

    Homework Statement Evaluate integral sin^(-3/2)(x)cos^3(x) dx Homework Equations tan(x)=sin(x)/cos(x) sin^2(x)+cos^2(x)=1 sin^2(x)=(1-cos(x))/2 cos^2(x)=(1+cos(2x))/2 integral cos(x)dx = sin(x) + c integral sin(x)dx = -cos(x) + c d/dx sin(x) = cos(x) d/dx cos(x) = -sin(x) a^m/a^n=a^(m-n)...
  6. G

    Calculus II - Trigonometric Integrals

    Homework Statement Evaluate integral( sin^3(x) cos^5(x) ) dxHomework Equations sin^2(x) + cos^2(x) = 1 integral x^n dx = x^(n+1)/(n+1) + c d/dx cos(x) = -sin(x) a^n*a^m=a^(n+m) The Attempt at a Solution I got -cos^6(x)/6+cos^8(x)/8+c Apparently I did something wrong SEE ATTACHMENT Thank...
  7. G

    Calculus II - Trigonometric Integrals

    Homework Statement Apparently I'm doing something wrong. I'm kind of lost as to what because I looked over my work several times. Homework Equations sin^2 x = ( 1 - cos 2x )/2 cos^2 x = ( 1 - sin 2x )/2 integral sin(x) dx = -cos(x) integral cos(x) dx = sin(x) The Attempt at a...
  8. Saitama

    Solving a Trigonometric Equation: Sec Theta

    Homework Statement This is a question from my test paper. If sec \theta=\frac{4xy}{(x+y)^2}, where x,y \in R, then (a)x,y \in R+ (b)x,y \in R- (c)x=y\neq0 (d)x\inR+, y\inR- Homework Equations The Attempt at a Solution I don't understand what i have to do and where should...
  9. S

    Trigonometric / Mixed Integral Involving Substitution

    Homework Statement integrate xarctanx/(x^2 + 1)^2 Homework Equations Integration by parts possibly? I was attempting to do it without integration by parts because we went over this in tutorial and my TA couldn't solve it properly, hence why I'm here. The Attempt at a Solution I...
  10. V

    Understanding Trigonometric Inverse Functions - Solving Homework Equations

    Homework Statement I understand that y = sin -1 x. However, why is it that when one writes x = sin y, one leaves out the -1 Homework Equations The Attempt at a Solution I know that when you invert a trig function, the dependent variable essentially switches from y to x and the...
  11. P

    Trigonometric Integration, Definite integral.

    Homework Statement Use 2 or more substitutions to find the following integrals hint : begin with u=cosx Homework Equations Integral 0--->pi/2 (cosxsinx)/swrt(cos2x+49 dx The Attempt at a Solution I'm still a little fuzzy on using multiple substitutions. From what I've read on...
  12. S

    Trigonometric simplification: -a*sin(wt) + w*cos(wt)

    I'm trying to figure out the steps required to do the following simplification: This equation: b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t)) can be simplified to the following: (b/√(a^2+ ω^2 ))*sin(ω*t+ θ) θ= tan^(-1) (- ω/a) I can numerically verify that this is true but I am having...
  13. J

    Derivatives of trigonometric functions - Question

    Homework Statement Find the Derivative of: (ln(cos4x)) / 12x^2 Homework Equations y' ln(x) = 1/x The Attempt at a Solution I have determined the correct answer, but I am still confused as to how I came to the solution. Starting with the numerator, the derivative of cos...
  14. nukeman

    Having Trouble: Solving a Trigonometric Equation using Identities ?

    Having Trouble: Solving a Trigonometric Equation using Identities ? Homework Statement I am having problems with these. Need some help please! Here is a sample question I am having troubles with: 3 cos θ + 3 = 2 sin^2 θ, 0 <= θ < 2pi I am really struggling with these. Any help...
  15. icesalmon

    Trigonometric Integral: Solving \int(cosx)^5sin(x)dx

    Homework Statement \int(cosx)^5sin(x)dx, the question asks me to simply find the integral. Homework Equations cos^2(x) + sin^2(x) = 1 The Attempt at a Solution I have an odd power of cosine so I should split that up into [cos^2(x)]^2(sin(x)cos(x)), then I change cos^2(x) to...
  16. K

    Does anybody know what came first for trigonometric functions?

    I want a timeline for trigonometric function and how it was developed. Just like who created it and stuff like that. Did the greeks know about trigonometric functions? Is a tangent in calculus the same as a tangent in trigonometry and the function os a trigonometry? How do trigonometric...
  17. R

    Two unknowns in trigonometric calculation?

    Homework Statement A boat leaves the harbor going 60 degrees north-east with the speed 20km/h. 3 hours later, another boat leaves a harbor 160km to the east with a speed of 70km/h. What course should the second boat have to meet the first? Homework Equations Not 100% sure what equations would...
  18. B

    Help proving trigonometric identity

    Homework Statement cos\theta/1-tan\theta+sin\theta/1-cot\theta=sin\theta+cos\theta Homework Equations The Attempt at a Solution
  19. A

    Why do trigonometric graphs with a π don't take inputs in radians/degrees?

    e.g. I want to find y when x is at pi/2 for the graph y=cos pi x. Why does pi/2 have to be expressed as a 'normal' (for want of better word) number (3.14/2) and not as 90 degrees (180/2) like you would for a graph without a pi in it e.g. cos x For cos x, if x is at pi/2, then I just put in...
  20. I

    Can following trigonometric equation be simplified:

    Fx = sin [ 90 - theta ] * [ dX - [tan (theta) * dY ] ] ??
  21. P

    How can I solve a trigonometric equation using algebra?

    Hi, I was asked to solve for this: sin (2x) = sin (0.5x) For 0 to 270 degrees I know how to solve it by plotting the two graphs, is there any other way of doing it, like, for example, algebraically? Thanks, Peter G.
  22. E

    How Do You Solve Complex Trigonometric Problems?

    1. Hi, I have a pre-calculus final tomorrow, and there are a few questions I don't understand. I'd truly appreciate it if you could help :) 1. If sin(theta)=1/4, theta in quadrant II, find the exact value of cos(theta+pi/6) 2. sin(sin^-1(2/3) + cos^-1(1/3)) -- simplify 3...
  23. P

    How can I find angles that satisfy 3sin(2x) = sin(x)?

    Hi, 3sin(2x) = sin (x) I managed to find two answers: 80.4 Degrees and 279.6 degrees but I don't know hot to get 0, 180 and 360 as answers, can anyone help me? This is how I found the two other angles: 6 sin (x) * cos (x) = sin (x) cos (x) = sin (x) / 6 sin (x) cos (x) = 1/6 Thanks in...
  24. P

    And the question is: How do I solve trigonometric equations?Hope this helps!

    Hi :smile: So, this is what I have to solve: 2sin2x - sin x = 0 I got two possibilities: sin x = 0.5 sin x = 0 From the second equation: I got 0 and 180, or, in other words, 0 and 3.14. The book has these two answers but the only other answer it has is 6.28 (and by the way, my domain is 0...
  25. P

    How Do You Determine the Minimum Length of a String in Trigonometric Motion?

    Hi :smile: A weight hangs on the end of an elastic string. The weight is pulled down and released from rest at time t = 0. The length x centimeters of the string at time t seconds is given by the formula: x = 7 - 2sin(πt) a) Sketch the graph of x against t for 0 <= x <=4 b) Find the two...
  26. P

    Trigonometric Formulas, Identities and Equation

    Homework Statement t = tetha FInd sin 2t when cos t = 4/5 Homework Equations no idea The Attempt at a Solution no. . . . btw what should I do 1st? I don't understand this question
  27. R

    Inverse Hyperbolic Funcs. or Trigonometric Substitution?

    So, in integrals that lead to inverse hyperbolics and can be solved with trigonometric substitution i just get lost. I know how to use both of them but i don't know which to use. For the sake of simplicity i'll just go with this one ∫ dx/sqrt(4+x^2) We know this equals arcsinh(x/2)...
  28. S

    ODE with trigonometric functions of solutions

    Dear all, Homework Statement Draw behavior around (0,0) of solutions to the following nonlinear system \left( \begin{array}{c} x'(t) \\ y'(t) \end{array}\right) =\left( \begin{array}{cc} cos {x(t)} + sin {x(t)} + {x(t)}^2 + {x(t)}^2{y(t)}^3 \\ -x(t) + {y(t)}^2 + y(t) + sin {y(t)}...
  29. O

    Value of 3 reciprocal trigonometric ratios

    Homework Statement For the angle shown a point is given on its terminal arm. Calculate the values of the three reciprocal trigonometric ratios, accurate to 4 decimal places if necessary. below is the link of the picture referring to the question https://www.virtualhigh...
  30. B

    Trigonometric Equation Homework Help: Solving 3sinx + 4cosx = 1

    Homework Statement I took a test in my Precalculus class about 3 weeks ago, and the tests were handed back this week. I saw that I didn't do so well because I was stumped on this equation: 3sinx + 4cosx = 1 Homework Equations We were to use the following formulas to solve this equation...
  31. S

    Confirming Trigonometric Equation Solutions

    i have solved some questions but my answers don't match with the solution list i have in my book... please if you help me check my answers, i'll be very grateful.. √3 tan x - sec x - 1 = 0 ... i get three solutions to this question but my book's got two... it doesn't mention 5pi/3 + 2n*pi...
  32. S

    Are My Trigonometric Equation Solutions Correct?

    solve the trigonometric equations tan 2(theta) + cot(theta) = 0 When i solved this question, i got pi/2 + 2n*pi and 3*pi/2 + 2n*pi ... but my book says the answer should be pi/6 + n*pi and 5pi/6 + n*pi Is my answer wrong?? if so, then what's your solution.?
  33. L

    Trigonometric functions problem

    Homework Statement Find x base of triangle: 2 pieces, 8+12 opposite end: x angle: \Theta ; angle formed by triangle of 8 units and x : 2\Theta Homework Equations tan\Theta = opp/adj tan 2\Theta = 2tan/1-tan2 The Attempt at a Solution tan\Theta= x/20 tan 2\Theta = x/8...
  34. B

    Caunchy's Integral Formula w/ trigonometric pole

    Homework Statement Solve the contour integral \int_{\gamma}\frac{2+z}{1-cos\left(z\right)}dz along a counter-clockwise circle of radius 1 centered at z=6 Homework Equations Cauchy's integral formula f\left(z\right)=\frac{1}{i*2\pi}\int_{\gamma}\frac{f\left(w\right)}{w-z}dw The...
  35. T

    Problem with finding multiple solutions to a trigonometric equation

    Homework Statement Find all solutions to the following equation: sin((5*x/2)+15)=0.433 where 0 < x < 360 Homework Equations The Attempt at a Solution i can find the first value of x that satisfies this equation. (5*x/2)+15=sin^-1(0.433) 5*x/2=10.6581 5*x=21.2162 x=4.2632...
  36. N

    Trigonometric root integral

    \intop_{-\pi/2}^{+\pi/2}\sqrt{(-6sin2t)^2+\sqrt{6}(cost)^2}dt how to solve it? i tried by parts and it looks very bad and complicated edit sorry i copied the wrond integral now it fine
  37. R

    How Do You Integrate \(\frac{\cos^2 x}{(1+\epsilon\cos x)^3}\) in Trigonometry?

    \int \frac{\cos^2 x}{(1+\epsilon\cos x)^3}\,dx Where, \epsilon > 0 is a real number constant.
  38. C

    Solving Trigonometric Equations: TanΘ=4/3

    Homework Statement TanΘ = 4/3 pi < Θ < 3pi/2 Find... a.) Sin(2Θ) b.) cos(2Θ) c.) Sin Θ/2 d.) Cos Θ/s The Attempt at a Solution So... The line will be in quadrant III. Would sin = 4 and cos = 3? What I used for a.) is ... a.) Sin(2Θ) = 2sinΘcosΘ =...
  39. C

    Solving Trigonometric Equations on Mobile

    I'm on mobile so I can't use latex.. sin^2 theta - cos^2 theta is not the same thing as one? If I had a problem say... 1 + cos^2 theta would that equal to sin^2 theta? Or -sin^2 theta? Thanks
  40. B

    Complex Trigonometric Solutions

    Homework Statement 2.) Solve x7=1 I am to solve for all the solutions to this polynomial, including the complex solutions using trigonometry. I'm supposed to plot my solutions on the unit circle, with the x-axis being the real axis, and the y-axis being the imaginary (or complex) axis...
  41. C

    Solving for theta in a Trigonometric Equation

    Homework Statement Tan \Theta = 12 / 5, sin\Theta<0Homework Equations Sin ^{2} \Theta+ Cos ^{2}\Theta = 1 The Attempt at a Solution Sin is less than 0 so... it should be somwhere in the II or III quadrant? What identities should I use to solve?
  42. Z

    How many times does cos(96πt)cos(4πt)=0 during t=0 to t=1s?

    Homework Statement This problem came up while solving a physics problem in waves. We have the equation cos(96πt)cos(4πt)=0 How many times does the L.H.S. become 0 during the time t=0 to t=1s ? The Attempt at a Solution Nothing.
  43. M

    Trigonometric Integration: Understanding the Magic Behind Integration Techniques

    This is more of a trig question but its coming out of an integration problem I am dealing with... the initial problem is... \int\frac{\sqrt(1+x^2)}{x}dx I've worked through the problem with the book and down the line I have to integrate csc(theta) without using a handy-dandy table...
  44. 1

    Trigonometric Equations find all solutions

    Homework Statement sec 3\Theta = 2 Find all solutions within 0 < \Theta < 2\Pi Homework Equations none The Attempt at a Solution I am having trouble "knowing" what angles satisfy sec = 2 within the interval. I can find one... ArcCos 1/2 = 1.047... which I can guess is...
  45. N

    System Of Trigonometric Equations

    a*sin(x)=b*sin(th)+c*sin(y) d+a*cos(x)=b*cos(th)+c*cos(y) Find x and y if all others are constants. I don't get how to solve this system of trigonometric equations in matlab. Please help me out friends. Thanks in advance.
  46. E

    Proving limit of trigonometric functions

    Homework Statement Prove: lim (n→0) {(sin n cos n) / (ⁿ√(1 - tan² 2n + tan⁴ 2n - tan⁶ 2n + ...))} = 1Homework Equations The Attempt at a Solution The denominator is an infinite geometric series, using the sum formula of an infinite geometric series, I simplify the limit: lim (n→0) {(sin n cos...
  47. F

    Trigonometric Limits: Find xsin(1/(x^2)) Limit as x->0

    Homework Statement Find the limit as x approaches 0 of xsin(1/(x^2)) Homework Equations The Attempt at a Solution I think this goes to 0, because the sine component just oscillates between 1 and -1, and gets multiplied by 0, for all x. I don't know how to show this using limit...
  48. L

    Prove Trigonometric Statement: Ideas & Solutions

    Homework Statement Prove http://img847.imageshack.us/img847/9428/trig.png Uploaded with ImageShack.us Homework Equations I am not quite sure how to prove this, I can see it involves using a double angle formula (see http://mathworld.wolfram.com/Double-AngleFormulas.html)...
  49. L

    Proving Trigonometric Identity: Double Angle Formula

    I am not quite sure how to prove this, I can see it involves using a double angle formula (see http://mathworld.wolfram.com/Double-AngleFormulas.html). I have tried working backwards but to no avail. Any ideas? http://img847.imageshack.us/img847/9428/trig.png Uploaded with...
  50. M

    Solving a Matrix Problem with Trigonometric Functions

    Hi, everyone, I am mathstkk, I am new to the Physics Forum, but I think, at my first sense, this forum is going to be helpful to me^^ Recently, I met one problem about matrix. The problem is as follow: Show that the matrix below has NO non-trivial solution if a+b+c=0 The matrix is 1...
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