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. archaic

    Trigonometric definite integral of 1/(4-sqrt(x))

    This could be solved by the substitution ##u=\sqrt x##, but I wanted to do it using a trigonometric one. The answer is false, but I don't see the wrong step. Thank you for your time! [Poster has been reminded to learn to post their work using LaTeX]
  2. R

    Exploring Trigonometric Functions & Physics: Velocity, Distance, & Dimension

    Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...
  3. V

    Solve for x with a trigonometric function

    Homework Statement: Solve for x. Homework Equations: sin(3x)= -1/2 sin(3x) = -1/2 3x = sin-1(-1/2) 3x = -π/6 x = -π/18 x = -π/18 + 2π/3 = 11π/18 11π/18 + 2π/3 = 23π/18 11π/18 + (2π(4))/3 = 35π/18 The solutions I obtained were 23π/18 and 35π/18. Are these correct? I'm not entirely sure...
  4. lfdahl

    MHB Trigonometric Sum Challenge Σtan^(-1)(1/(n^2+n+1)=π/2

    Show that \[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\] - and deduce that \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]
  5. D

    Solving a Trigonometric Equation: v^2*sin(180-2theta2)/g

    theta1 = 90- theta2 I substituted that into v^2*sin(2theta1)/g So I get v^2*sin(180-2theta2)/g Now I'm stuck. What do I do next?
  6. J

    A Creation/annihilation operators and trigonometric functions

    Hello everyone, I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...
  7. P

    MHB Unraveling a Trigonometric Mystery

    We know the answer, but don't know how it makes sense given trigonometric principles. Borrowed from HiSet free practice test
  8. Y

    MHB Trigonometric Question sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x)

    In this question, I tried this: sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x), where cosec(x) = 1/sin(x) and sec(x) = 1/cos(x) -sin^2(180-x) = sin^2(x) and cos^2(x) = cos^2(x) -The sin^2 and the 1/sin(x) cancle out along with the cos^2 and the 1/cos(x) Therefore, I am left with...
  9. M

    An integration problem using trigonometric substitution

    This is the integral I try to take. ##\int\sqrt{1+9y^2}## and ##9y^2=tan^2\theta## so the integral becomes ##\int\sqrt{1+tan^2\theta}=\sqrt {sec^2\theta}##. Now I willl calculate dy. ## tan\theta=3y ## and ##y=\frac {tan\theta}3## and ##dy=\frac{1+tan^2\theta}3## Here is where I can only...
  10. babaliaris

    Uniform Circular Motion: some help with the math proof?

    I can not understand why ##v_x = -|v|sin(θ)## and ##v_y = |v|cos(θ)## I'm asking about the θ angle. If i move the vector v with my mind to the origin i get that the angle between x'x and the vector in anti clock wise, it's 90+θ not just θ. So why is he using just θ? Does the minus in v_x somehow...
  11. J

    Solving a system of two simultaneous trigonometric equations

    Homework Statement I need to solve a system of two equations for T and θ algebraic and with all the other parameters known. φ is equal to: Homework Equations The relevant equations are the two equations left of * in the image below The Attempt at a Solution I tried Gauss elimination but I...
  12. QuarkDecay

    Solving a trigonometric equation for the angle

    Homework Statement The equation; 0.966= -0.354sin(φ+60) + 0.935cos(φ+60) and we're trying to find φ. Homework Equations 3. The Attempt at a Solution [/B] (edited) I tried doing ^2; 0.933= 0.125sin2(φ+60) -2*0.331sin(φ+60)cos(φ+60) + 0.874cos2(φ+60) x=φ+60 0.933= 0.125sin2x - 0.331sin(2x) +...
  13. C

    MHB Trigonometric Riddle: Determine Wind Speed & Direction

    Hi, I got a problem I am trying to solve. An airplane flies at a fixed air speed which is unknown. There is a wind with unknown heading and unknown speed. The airplane has a known ground speed and direction. The airplane changes heading (with reference to ground), and now there is a different...
  14. S

    B Are trigonometric ratios physical quantities?

    I already know the fact that angles are physical quantities, but sin, cos of some angles are quantities? Quantities are those things, which can be quantified, are sin, cos, tan be quantified through measurement, if yes then other mathematical functions should also be categorised as physical...
  15. physicophysiology

    Trigonometric function (Mechanics-Landau)

    <Moderator's note: Moved from a technical forum and thus no template.> Mechanics by Lev D. Landau & E. M. Lifshitz Chapter 4 Collisions between particles §16. Disintegration of particles Problem 3 The angle θ = θ1 + θ2 It is simplest to calculate the tangent of θ. A consideration of the...
  16. L

    B What values of m as a function of q satisfy this trigonometric equation?

    I have a trigonometric equation 2\sin \left ( \frac{q\pi }{m} \right )-\sin \left ( \frac{q\pi }{2} \right )=0 and want to know what values m as a function of q could take to satisfy the equation. Both terms zero is the obvious solution: q=2n; m=2; n is an integer. But there are more solutions...
  17. J

    MHB Derivatives of trigonometric functions

    The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain! - - - Updated - - - Oh nvm I see my mistake!
  18. anemone

    MHB Trigonometric inequality: sin (1/(n+1934))<1/1994

    Find the smallest natural number $n$ for which $\sin \left(\dfrac{1}{n+1934}\right)<\dfrac{1}{1994}$.
  19. Krushnaraj Pandya

    Area under an inverse trigonometric function

    Homework Statement Find the area bounded by arcsinx, arccosx and the x axis. Hint-you don't need to integrate arcsinx and arccosx Homework Equations All pertaining to calculus The Attempt at a Solution I drew the correct graph and marked their intersection at (1/√2, pi/4) and painstakingly...
  20. Krushnaraj Pandya

    Proof of an inverse trigonometric identity

    Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...
  21. Krushnaraj Pandya

    Definite trigonometric integral

    Homework Statement solve ##\int_0^1 x^6 \arcsin{x} dx##
  22. Krushnaraj Pandya

    Definite trigonometric integral using properties

    Homework Statement If ## I_n = \int_0^\frac {\pi}{4} \sec^n x dx## then find ## I_{10} - \frac {8}{9} I_8## 2. The attempt at a solution this should be solvable by reduction formulae but since it'd be longer I wanted to know if there was a way to do it using mostly properties of indefinite...
  23. Krushnaraj Pandya

    Simple trigonometric simplification

    Homework Statement Show that sec(x-(pi/2))+tan(x-(pi/2))=tan((x/2)+pi/12)) The Attempt at a Solution I applied all sorts of half angle formulas to convert it in terms of tan, I got LHS as (tan((x/2)-(pi/6))+1)^2/1-tan^2(x/2-(pi/6)) but I'm sure there must be a simple easy method to get the RHS...
  24. Bobs

    Today, in our class, we received a trigonometric equation

    Today,in our class, we received a trigonometric equation ##\sin^{10}{x}+\cos^{10}{x}=\frac{29}{16}\cos^4{2x}## Here is my attempt:
  25. S

    MHB How to Prove a Trigonometric Identity Involving x, y, and z?

    If xy+yz+zx=1 then prove (x^2-1)(y^2-1)/xy+(y^2-1)(z^2-1)/yz+(z^2-1)(x^2-1)/zx=4 with trigonometric identities such as cotAcotB+cotBcotC+cotCcotA=1
  26. opus

    Isolating the Y variable in a Trigonometric Equation

    Homework Statement ##4y=cos\left(4πx+\frac{3}{2}\right)## Homework EquationsThe Attempt at a Solution In dividing both sides by 4, I got: ##y=\frac{1}{4}cos\left(πx+\frac{3}{8}\right)## But I am told this is incorrect. Not sure if dividing everything by 4 here is an allowable technique, or if...
  27. D

    I Name those trigonometric functions

    The circle ## x^n+y^n=1 ##, for n integer >2 in a metric space with distance function: ## \sqrt[n] {dx^n+dy^n} ## has corresponding trigonometric Sine and Cosine functions defined in the usual way. Finding the sine or cosine of the sum of two angles, derivatives and curvature of a line in such...
  28. E

    A How to Approach Solving a Complex Trigonometric Integral?

    Hello everyone Can someone help me out solving this integral: \begin{equation} S_T(\omega)=\frac{2k_BT^2g}{4\pi^2c^2}\int_0^{\infty}\frac{sin^2(kl)}{k^2l^2}\frac{k^2}{D^2k^4+\omega^2}dk \end{equation} Where $$D=g/c$$ According to this paper https://doi.org/10.1103/PhysRevB.13.556. The...
  29. Amlan mihir

    Find the value of the trigonometric sum

    Homework Statement i have no idea how to proceed
  30. Physiona

    Angles involving trigonometric Ratios Worded Problem

    Homework Statement Triangle ABC is shown in the diagram below. A C AC = 3AB <BAC = 120° Respectively <BCA = Show that angle BCA can be written in the form...
  31. J

    MHB Prove Trigonometric Identity

  32. lfdahl

    MHB Trigonometric product challenge

    Prove, that $$\prod_{j = 1}^{n}\left(1+2\cos \left(\frac{3^j}{3^n+1}2\pi\right)\right) = 1.$$
  33. M

    MHB Evaluating Trigonometric Expressions

    Given h(x) = tan x, evaluate dh/dx on [pi/4, 1]. Note: d = delta I need one or two hints. I can then try on my own.
  34. M

    MHB How Can I Evaluate Trig Functions Without a Calculator?

    I am in the trigonometry section of my precalculus textbook by David Cohen. In Section 6.2, David explains how to evaluate trig functions without using a calculator but it is not clear to me. Sample: Is cos 3 positive or negative? How do I determine if cos 3 is positive or negative without...
  35. Y

    MHB Trigonometric Integral, weird results

    Hello all, I am trying to solve the integral: \[\int cot(x)\cdot csc^{2}(x)\cdot dx\] If I use a substitution of u=cot(x), I get \[-\frac{1}{2}cot^{2}(x)+C\] which is the correct answer in the book, however, if I do this: \[\int \frac{cos(x)}{sin^{3}(x)}dx\] I get, using a substitution...
  36. Monoxdifly

    MHB [ASK] Trigonometric Equation

    If 0 < x < 90, what is the solution of tan(2x - 5) = cot(x + 5)? I got stuck in tan(2x - 5)tan(x + 5) = 1. What should I do after that?
  37. H

    What Are the Solutions for cos(2x) + cos(x) = 0?

    Homework Statement cos2x + cos x = 0 (0 <= x <= 360) Homework EquationsThe Attempt at a Solution cos2x + cos x = 0 2cos(3x)/2 cos(x)/2 = 0 3x/2 = 90 degrees x = 60 degrees x/2 = 90 x = 180 3x/2 = 270 x = 180 x/2 = 270 x = 540 (not qualified) is there any more possibility (answers) for x?
  38. Adgorn

    Need help with a trigonometric expression

    Homework Statement Just a simple problem, I need to take the expression##\frac 1 2 (sin(2x)+1)## and show it is equivalent to ##sin^2(x+\frac \pi 4)##, and I can't seem to manage to find the way to do so, so I would appreciate some insight. Homework Equations N/A The Attempt at a Solution...
  39. lfdahl

    MHB What is the Proof for the Trigonometric Sum Identity?

    Prove the identity \[\sum_{j=1}^{n-1}\csc^2\left ( \frac{j\pi}{n} \right ) = \frac{n^2-1}{3 }.\]
  40. J

    How Do You Simplify Trigonometric Expressions Using Basic Identities?

    Homework Statement Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx Homework Equations cot(x) = 1/tan(x) sin^2(x) + cos^2(x) = 1 The Attempt at a Solution I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved...
  41. E

    MHB Derivatives of trigonometric equation

    Could I please get help with the following question? f(x)=(2cos^2 x+3)^5/2 Any help would be very much appreciated:)
  42. karush

    MHB T1.14 Integral: trigonometric u-substitution

    $\tiny{2214.t1.14}$ $\text{Evaluate the Integral:}$ \begin{align*}\displaystyle I_{14}&=\int \frac{12\tan^2x \sec^2 x}{(4+\tan^3x)^2} \, dx \\ \textit{Use U substitution}&\\ u&=4+\tan^3x\\ \, \therefore dx& =\dfrac{1}{3\sec^2\left(x\right)\tan^2\left(x\right)}\,du\\ &=4...
  43. Greg

    MHB Trigonometric Sum Prove: N=3,5,7...

    Prove $$\sum^{(N-1)/2}_{n=1}\cos\left[\frac{\pi}{N}(2n-1)\right]=\frac12$$ For $N=3,5,7...$.
  44. anemone

    MHB Trigonometric equality sin15°sin24°sin57°=sin39°sin27°sin18°

    Prove $\sin 15^\circ \sin 24^\circ \sin 57^\circ= \sin 39^\circ \sin 27^\circ \sin 18^\circ$. This is an unsolved problem I found @ AOPS.
  45. R

    MHB Stuck on a trigonometric identity proof....

    $\frac{1 -\cos A}{1 + \cos A} = (\cot A - \csc A)^2$
  46. H

    B Eliminating Variables in Trigonometric Equations for Research Purposes

    Consider the following set of equations: ##r = \cosh\rho \cos\tau + \sinh\rho \cos\varphi## ##rt = \cosh\rho \sin\tau## ##rl\phi = \sinh\rho \sin\varphi## Is there some way to combine the equations to get rid of ##\varphi## and ##\tau## and express ##\rho## in terms of ##r, t, \phi##? I...
  47. A

    Trigonometric substitution, What am i doing wrong?

    Homework Statement Homework Equations The Attempt at a Solution Here is my answer, i get 1/24 For my first step i divided both terms under the radical by 4, then split 1/4 into (1/2)2, i saw something very similar in my book so i did the same thing, but i just realized this has to be...
  48. lfdahl

    MHB Solve Trigonometric Inequality 5x≤8sinx−sin2x≤6x

    Show, that $5x \le 8\sin x - \sin 2x \le 6x$ for $0 \le x \le \frac{\pi}{3}$.
  49. DeathbyGreen

    I Infinite series of trigonometric terms

    I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...
  50. lfdahl

    MHB Evaluate the limit of a trigonometric expression

    Evaluate: \[\lim_{x\rightarrow 0}\frac{\sin (\arctan x)-\tan (\arcsin x)}{\arcsin (\tan x)-\arctan (\sin x)}\]
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