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.

View More On Wikipedia.org
Recent contents Top users
  1. M

    Trigonometric Identities algebra

    hello. i have a question about trignometric identities.. it's realtivly easy, but am struggling with the algebra in it ( Algebra + trig = Very confusing to me ) Prove that .. [Sinx/(1+Cosx)] + [(1+cosx) / sinx] = 2csc x i manged to get it to [Sin2x+1+2cosx+cos2x] / SinxCosx...
  2. D

    Integral of trigonometric functions

    Homework Statement How can I integrate this: \int sin (nt) sin (n \pi t) dt This actually in the Fourier series.
  3. L

    Trigonometric Integral sanity check

    Homework Statement Hello, After some algebraic manipulation my arc length formula of sqrt(1+y')^2) resulted in sqrt(1+ tan^2 (x))...which can translate into sqrt(sec^2(x)). My question is does the square root cancel the sec^2 ? becoming sec x? Like sqrt(x^2) is x? Thanks...
  4. T

    2 Trigonometric Problems i'm stuck with. Help

    Hi. I would appreciate if anybody could help me with the following: 1. a)Solve 3cosec x-5sin x = 2 for 0<x<360degrees b) Prove that cosecx - sinx = cosx cotx 2 a) f(x) is x + ln(3x-4). Show how to convert f(x) into the iterative formula b) If x0=1, write values of x1, x2, 3 until a...
  5. S

    Arc length of vector function with trigonometric components

    Homework Statement Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified in each case. r(t) = (c2/a)cos3t i + (c2/b)sin3t j where i and j are the usual unit vectors, 0 \leq t \leq 2\pi, c2 = a2 - b2, and 0...
  6. J

    Limit with trigonometric functions

    Homework Statement True or false? lim Tanx/(1-Cosx) = lim Sec2x/Sinx = Infinity (limits are as x approaches \pi from the left) The Attempt at a Solution I tried just plugging \pi into x in the first limit, and I ended up getting 0/2, which exists but is just 0. So...
  7. H

    Calculus BC Trigonometric Substitition

    Homework Statement \int \frac {\sqrt {1+x^2}}{x} \, dx Homework Equations \sin^2{\theta}+\cos^2{\theta}=1 The Attempt at a Solution x=\tan{\theta} I simplified it down to: \int \frac {1}{\sin{\theta} \cdot \cos^2{\theta}} \, d\theta Which I do not know how to integrate. Any help would...
  8. A

    Find the limit of trigonometric function

    hi there, just new to this form
  9. T

    Integration with inverse trigonometric substitution

    Homework Statement The problem is: integrate: (1/(4x2+4x+5)2)dx Homework Equations The Attempt at a Solution 4x2+4x+5=(2x+1)2+4 gives: ∫dx/((2x+1)2+4)2 use regular substitution: u=2x+1 du=2dx dx=1/2du gives: 1/2∫du/(u2+4)2) trigonometric substitution: u=tan(z)...
  10. Z

    Link between Pascal's triangle and integrals of trigonometric functions

    My lecturer keeps simplifying trigonometric integrals in one line such as \int^{2\pi}_{0}sin^{4}(t)dt=\frac{3\pi}{4} and writes pascals triangle next to it. Just wondering what's the link between them? I'm sure it's obvious and easy, I'd just like to have an fast way of dealing with these
  11. I

    Trigonometric derivation question

    Homework Statement Determine all points on the function y=\sin2x-2sinx where the tangent is parallel to the x-axis.Homework Equations f'(sin(x)) = cos(x) chain rule: f'(f(g(x)) = g'f + f'g The Attempt at a Solution y=sin2x-2sinx y\prime=cos(2x)(2)-[2cosx+0sinx] y\prime=cos(2x)(2)-2cosx If...
  12. V

    Limit of a trigonometric function (Involved problem)

    Homework Statement [B]Evaluate \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \frac{x}{2}-1}{x\sin x} , WITHOUT using l'Hôpital's rule. Homework Equations The Attempt at a Solution Hello there, I tried to evaluate this limit using two different approaches, both of which still...
  13. V

    Limit of a trigonometric function (Advanced problem)

    Adminstrator: This is a double post, so please feel free to delete this one. The relevant post is "Limit of a Trigonometric Function (Involved Problem)". Homework Statement Evaluate \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \frac{x}{2}-1}{x\sin x} Homework Equations The Attempt at a...
  14. G

    How to Integrate cos4 x Using the Double Angle Formula

    Homework Statement Find the integral of cos4 xdxHomework Equations cos2x = 2cos2x-1 cos2x = (cos2x+1)/2The Attempt at a Solution I tried using cos4x = cos2x * cos2x and i simplified it to cos4x = ((cos2x)2 + 2cos2x + 1)/4 I'm not sure if this is right. How do i go about getting the integral...
  15. M

    Trigonometric Integral Homework: Solving \int (sin^6(x))(cos^3(x))

    Homework Statement \int 41(sin^{6}(x))(cos^{3}(x)) Homework Equations The Attempt at a Solution I think you are supposed to use the half angle identities and then maybe integration by parts but I am lost on it.
  16. K

    Calculus 2: Trigonometric Substitution, using Z = tan(x/2)

    Homework Statement Homework Equations The Attempt at a Solution After substituting: Using I'm stuck here: I can't seem to find anything online about this substitution. Any help would be appreciated. thanks.
  17. T

    Another Trigonometric Equation

    Homework Statement Solve for t sin3tcost+cos3tsint=-1/2 Homework Equations Sum and Difference Identities The Attempt at a Solution I saw this problem the the sum identity sin(\alpha+\beta) looked like the way to begin solving for t. This condenses to sin(3t+t)= -1/2 which...
  18. S

    When does a real solution exist for this system of trigonometric equations?

    So, my question is really simple. What are the requirements for a1, a2, C1, C2 for a real solution to exist for the following system of equations? sin(theta1) + a1*sin(theta2) = C1 cos(theta1) + a2*cos(theta2) = C2 If it would be helpful, I'm not interested in the solution, necessarily (could...
  19. B

    Trigonometric Identity and Differential Equation question

    I'm looking over the differential equation describing a hanging cable in a textbook, and I probably need to review my trigonometric derivatives and integrals again because I'm not seeing how they got the following: \frac{dy}{dx} = tan(\phi) \frac{ws}{T_0} \frac{d^2y}{dx^2} =...
  20. R

    Integrals of Trigonometric Functions

    I have three problems that I'm having a hard time with. I'd appreciate any help with any of the three problems. \int((cos(x))^6)dx AND \int(x^3 * sqrt(x^2 - 1) AND Solve for y (separation of variables): dy/dx = ((2y +...
  21. R

    Integrals of Trigonometric Functions

    Homework Statement I have three problems that I'm having a hard time with. I'd appreciate any help with any of the three problems. \int((cos(x))^6)dx AND \int(x^3 * sqrt(x^2 - 1) AND Solve for y (separation of...
  22. M

    How to Simplify a Trigonometric Limit and Solve for 1/2

    This is from Thomas' Calculus Early Trancendentals, Media Upgrade, page 108, #27: The limit as x approaches 0 of (x csc(2x))/cos(5x) The answer is 1/2, which graphing confirms, but hell if I know how to get rid of the 5x and still come up with 1/2. It is boggling my mind. Any help would be...
  23. L

    What is the Integral of a Trigonometric Function with a Weird Substitution?

    Homework Statement I need to calculate the integral: \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\mathrm{d}x Homework Equations The Attempt at a Solution There is a tip: "try substituting y=\frac{\pi}{2}-x. I tried it and didn't get anywhere. I also tried several...
  24. T

    Trigonometric derivatives and roots of unity

    sin x. d(sin x)/dx = cos x. d(cos x)/dx = -sin x. d(-sin x)/dx = - cos x. d(-cos x)/dx = sin x. i. i^2 = -1. i^3 = -i i^4 = 1 i^5 = i. I know there is a relationship between trig, the complex numbers, and exponential functions. Is there a relationship between the pattern shown here?
  25. T

    Simplifying Trigonometric Expressions

    Homework Statement I'm not sure if you can ask about two questions in one thread; they are very similar. If it's not allowed I can make a second thread and edit this one. Simplify the expressions: 1. sin^3 theta + cos^3 theta / sin theta + cos theta 2. 2-tan theta / 2 csc theta-sec...
  26. J

    Integration by Trigonometric Substitution

    Homework Statement Evaluate: \int\frac{1}{(4 - \tan^2{x})^{3/2}}dx Homework Equations I must integrate the above equation using only trigonometric subtitutions of algebraic equations. The Attempt at a Solution Here is what I have so far: Let \tan{(x)} = 2\sin{(\theta)} x =...
  27. W

    Trigonometric Identities Proof

    1. (A) sin(-x) = - sin x (C) cos(x+y) = cosxcosy - sinxsiny (B) cos(-x) = - cos x (D) sin(x+y) = sinxcosy + cosxsiny Use these equalities to derive the following trigonometric identities. a. absolute value of cos x/2 = \sqrt{}1 + cosx/2 b. absolute...
  28. A

    Solving simultaneous trigonometric equation

    how to solve eqn1 = sin((alpha)*a)*cosh((alpha)*b)=sin(alpha)*a eqn2 = sinh((alpha)*b)*cos((alpha)*a) = b*sin(alpha) alpha = 0 to 2pi... Find a & b... Plz help me with any method that would help solving this...
  29. drizzle

    Solving Trigonometric Equations with Excel (Vista)

    :blushing: using excel (vista) how can I reverse the number to be in degrees rather than radians if I want to use trigonometric functions? I have to reverse the number of the angle in radian to use those functions and I want them to be in degrees, anyone know how, thanks in advance.
  30. S

    Integrating cos^2(2x): Where Did I Go Wrong?

    Homework Statement \int cos^{2}(2x) dx Homework Equations The Attempt at a Solution After doing all of the steps I came out with \frac{x+sin4x}{8}+c In the text it says the answer is \frac{x}{2}+\frac{sin4x}{8}+c Where did I go wrong?
  31. J

    Limit of a trigonometric function

    Homework Statement Find the limit as x approaches 0 of f(x) = 3x / sin 5x Homework Equations I might have to use the fact that "the lim of x approaches 0 of sin x/x = 1" The Attempt at a Solution I don't know how to start to solve the expresion, I would appreciate any...
  32. G

    Trigonometric Substitution for Evaluating Integrals: Using t=atan(theta) Model

    Homework Statement Use an appropriate substitution and then a trigonometric substitution to evaluate the integral: \int_-^{ln(4)} \frac{e^{t}dt}{\sqrt{e^{2t}+9}} Homework Equations Seems to be close to the t=atan(\theta) model \int \frac{e^{t}}{\sqrt{a^{2}+x^{2}}}dt The Attempt at a...
  33. J

    Prove 1+1=2 using trigonometric functions

    Prove 1+1=2 using trigonometric functions
  34. Mentallic

    How do I solve the trigonometric integral of f(x)=cotx+tanx?

    Part 2, leading on from https://www.physicsforums.com/showthread.php?t=315803" Homework Statement Given f(x)=cotx+tanx find \int{[f(x)]^2}dx The Attempt at a Solution I've attempted many different varieties of approaches to the problem. Trying to use the substitution method for sinx...
  35. Mentallic

    Trigonometric Integral: Finding the Antiderivative of cotx + tanx

    Homework Statement If f(x)=cotx+tanx then find \int{f(x)}dx The Attempt at a Solution I was able to manipulate (not sure if in the right direction though) the function and resulted with f(x)=\frac{2}{sin(2x)} I've also tried re-arranging things in a different way, but came up with...
  36. P

    Trigonometric function with specific properties

    Is there any trigonometric function f(theta) where theta is an angle such that f(Pi-theta)=-f(theta) and such that its square (i.e f(theta)^2) is maximum at Pi/2? Thanks
  37. S

    To maximise a trigonometric function

    Homework Statement What is the maximal value of f(x) = sin(x) - \sqrt{3} cos(x) such that 0 <= x < 360? The Attempt at a Solution To raise to power to two f^2 = sin^2(x) - 2 \sqrt{3} sinx cosx + 3cos^2(x) I obverse that the last term has the greatest coeffient so I maximise...
  38. L

    Finding definite integral (trigonometric)

    Homework Statement Hi guys ,, i have the following question (it's in the attachment) : Find m and M such that m <= x sin x <= M if 0 <=x <= pi. (Any reasonably good bounds will do, I am not asking for the best possible bounds.) Hence find bounds on the value of the integral[x sin(x),0 to...
  39. N

    Trigonometric Function Homework: Solving sin^2 (x - pi/4) = 1

    Homework Statement Pardon me first of all for any mistakes, I study in a French school so I might use incorrect terms I have to find the solutions for the following formula: sin^2 ( x - pi/4 ) = 1 Homework Equations None? The Attempt at a Solution I've mostly only worked with...
  40. M

    Proving Trigonometric Identities

    Homework Statement cosx - cosy=-2 sin(x + y/2) sin(x - y/2) Homework Equations dont know what identities to use The Attempt at a Solution ok so when i figure it out, the RHS always comes out to either... (cos x sin y - sin x cos y)/2 or just cos x sin y - sin x cos y...
  41. M

    Proving Trigonometric Identities

    Homework Statement 1) sin(x+y)sin(x-y)=cos^2y-cos^2x 2) tan(∏⁄4+x)+tan(∏/4-x)=2sec2x 3) cosx-cosy=-2sin(x+y/2)sin(x+y/2) 4) 2cotx-2tanx=4-2sec^2x/tanxHomework Equations all trig identitiesThe Attempt at a Solution 1) i understand that i should show what i have attempted but there is way too...
  42. B

    Maximizing Power in an Electric Circuit: Trigonometric Problem Solution

    Can anyone please give me some idea for this problem? The instantaneous power, p, in an electric circuit is given by p = iv, where v is the voltage and i is the current. Calculate the maximum value of power in the circuit if v=0.02 sin(100 pi t)volts and i=0.6 sin(100pi t+ pi/4 ) amp...
  43. N

    Solving Trigonometric Equations?

    Homework Statement Solve sin^2(x) = cos^2(x) over the interval [0,2pi] The Attempt at a Solution sin^2(x) = cos^2(x) sin^2(x) = 1-sin^2(x) 2sin^2(x) = 1 sin^2(x) = 1/2 = pi/6 sin(x) = sq root 1/2 = sq root pi/6 Unfortunately I am completely lost at this point and am not sure if I...
  44. D

    Simplifying the Trigonometric Expression Question

    Homework Statement cosx/(secx+tanx) The Attempt at a Solution So far I have this: =cosx/(1/cosx + sinx/cosx) =cosx/(sinx + 1/cosx) =cos2x/sinx + 1 From there I don't know what to do. I've messed around with it for a while but don't seem to get anywhere. Any help is appreciated.
  45. ShayanJ

    The period of trigonometric functions

    Hi everyone Could you give me a way to calculate the period of every trigonometric functions? thanks
  46. X

    Solving a Trigonometric Equation: (cos x)^4 = cos (x^2)

    Hi all, I was tutoring someone in Algebra III today and encountered a problem that threw me for a loop. (cos x)^4 = cos (x^2) Is there a way to solve this algebraically?
  47. L

    Simplify trigonometric equation problem

    Homework Statement Simplify the following. Write all trigonometric functions in terms of t. Homework EquationsThe Attempt at a Solution I know that: sec(t+2pi)=sec(t) 1+tan(t+3\pi)= 1+tan(t) What about: csc(t-6\pi)? Will it be equal to csc(t). Can anyone please help me with this? Thank you...
  48. I

    Can the set of x be found for -1 ≤ 1/(1+cosx) ≤ 1?

    Homework Statement solve the inequality: -1\leq 1/(1+cosx) \leq1 The Attempt at a Solution firstly cosx does not equal -1 but you'll see doesn't help much -1<cosx<1 0<cosx+1<2 that makes the above equation 1/(1+cosx) between 0.5 and infinity. but I have no...
  49. L

    Finding Trigonometric Limits through Calculator Techniques

    Homework Statement Let f(x)=cos (x). Denote f^2(x) = ff(x), f^3(x)=fff(x) and so on. For x=100 radians, find f^n(x) for n=1 to 20 Develop a calculator technique to find f^n(x) for any given value of n. Hence, find correct to 7 decimal places, the limit of f^n(x) for x= 100 radians and n...
  50. 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...
Back
Top