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. anemone

    MHB Solution to Trigonometric System

    Find all reals x,\,y,\,z \in \left[0,\,\frac{\pi}{2}\right] that satisfying the system below: $\sin x \cos y=\sin z\\\cos x \sin y=\cos z$
  2. T

    MHB Integral of trigonometric function

    I have this integral to solve. $$\int_{}^{} (sinx + cos x)^2 \,dx$$ I first start by simplifying the expression: $$\int_{}^{} sin^2x + 2sinxcosx + cos^2x \,dx$$ $$2sinxcosx$$ is $$sin2x$$ (a trigonometric identity) and $$ sin^2x + cos^2x = 1 $$ a trigonometric identity. So, after...
  3. F

    Trigonometric equation for stress

    Homework Statement is the circled part wrong ? Homework EquationsThe Attempt at a Solution how could 0.5 sin2theta = ( 1 + cos2theta ) / 2 ?
  4. anemone

    MHB What is the ratio of sin 5x to sin x in this Trigonometric Challenge?

    Given that \frac{\sin 3x}{\sin x}=\frac{6}{5}, what is the ratio of \frac{\sin 5x}{\sin x}?
  5. H

    I Graphs of inverse trigonometric vs inverse hyperbolic functions

    I noticed the graphs of ##y=\cos^{-1}x## and ##y=\cosh^{-1}x## are similar in the sense that the real part of one is the imaginary part of the other. This is true except when ##x<-1## where the imaginary part of ##y=\cos^{-1}x## is negative but the real part of ##y=\cosh^{-1}x## is positive. I...
  6. 9

    Differentiate trigonometric equation

    Homework Statement a) Differentiate the following equation with respect to: 1) θ 2) Φ 3) ψ (Ua - Ub)' * C * r where: C is a 3 x 3 rotation matrix: [ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ] [ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin...
  7. RoboNerd

    I Question on basic trig substitution with x = sin theta

    Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused. Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta? Thanks in advance for...
  8. anemone

    MHB Can You Prove this Trigonometric Inequality Challenge?

    Let the real $x\in \left(0,\,\dfrac{\pi}{2}\right)$, prove that $\dfrac{\sin^3 x}{5}+\dfrac{\cos^3 x}{12}≥ \dfrac{1}{13}$.
  9. L

    B Simple question about differentiation of trigonometric function

    Explain to me: Why the 2πf came in front? I lost touch and sort of forgot.
  10. R

    B Transform the system of trigonometric equations

    How to extract l and L from the following system of equations:
  11. terryds

    Solving a Trigonometric Limit Problem

    Homework Statement ##\lim_{a\rightarrow b} \frac{tan\ a - tan\ b}{1+(1-\frac{a}{b})\ tan\ a\ tan\ b - \frac{a}{b}}## = ...Homework Equations tan (a - b) = (tan a - tan b)/(1+tan a tan b) The Attempt at a Solution [/B] I don't know how to convert it to the form of tan (a-b) since there are...
  12. Draconifors

    Quick Trigonometric Identity Question

    Hi! I have an integral to solve (that's not the point, though) and the inside of the integral is almost a trig identity: 1. Homework Statement ##sin\frac{(x+y)} {2}*cos\frac{(x-y)} {2} ## Homework Equations I noticed this was very similar to ##sinx+siny = 2sin \frac{(x+y)} {2} *...
  13. D

    Solution to this trigonometric equation

    Homework Statement ##tanx=\frac{(1+tan1)(1+tan2)-2}{(1-tan1)(1-tan2)-2}## find x Homework Equations 3. The Attempt at a Solution [/B] I tried multiplying through the paranthesis and arrived at ##tanx=\frac{(tan1tan2-1)+(tan2+tan1)}{(tan1tan2-1)-(tan2+tan1)}## and i don't know if this is any...
  14. prashant singh

    I Why trigonometric ratios were defined for a unit circle

    To make it useful for any angles. I need a good explanation for this.
  15. V

    What is the proof for 2sin2θ - 1 = sin2θ - cos2θ?

    Homework Statement 2sin2θ - 1 = sin2θ - cos2θ Homework EquationsThe Attempt at a Solution I am unsure of how to prove these. So far all I have is Left side= 2sin2θ - 1 =sin2sin2-1 And I know that right side is equal to 1. But otherwise not sure where to go from there.
  16. P

    Finding sum of roots of trigonometric equation

    Homework Statement Question: Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is: (a) 5050 π (b) 4950 π (c) 5151 π (d) none of these The correct answer is: (b) 4950 π Homework Equations ## cos(2x) = 2cos^2(x) -1 ## The Attempt...
  17. anemone

    MHB How do I prove a trigonometric inequality?

    Prove that for all real numbers $x$, we have \left(2^{\sin x}+2^{\cos x}\right)^2\ge2^{2-\sqrt{2}}.
  18. anemone

    MHB How to Prove the Trigonometric Inequality for Real Numbers?

    For real numbers 0\lt x\lt \frac{\pi}{2}, prove that $\cos^2 x \cot x+\sin^2 x \tan x\ge 1$.
  19. A

    Trigonometric equation sin(x) = C*sin(y)

    Homework Statement sin x = C*sin y Find y as a function of x for a given C>0. Homework Equations sin x = C*sin y The Attempt at a Solution This is not actually a problem from a book, but a problem I myself thought about. I was studying elastic collisions in SCM and I obtained 2 equations...
  20. anemone

    MHB Can This Trigonometric Inequality Be Proven for All Real Numbers?

    Prove that \frac{\sin^3 x}{(1+\sin^2 x)^2}+\frac{\cos^3 x}{(1+\cos^2 x)^2}\lt \frac{3\sqrt{3}}{16} holds for all real $x$.
  21. anemone

    MHB How can you maximize a trigonometric expression?

    Maximize $\sin x \cos y+\sin y \cos z+\sin z \cos x$ for all real $x,\,y$ and $z$.
  22. S

    Trying to prove trigonometric integrals on a quarter of circle

    Homework Statement I want to prove that: Homework EquationsThe Attempt at a Solution I tried using the trigonometric identity: sen2x = senx cosx / 2, so, I got: 1/2m∫(sen2x)mdx, x from 0 to pi/2, but now I don't know how to proceed. Can you help me please?
  23. T

    Proving trigonometric identities in a belt and pulley proble

    Homework Statement verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C] 2. The attempt at a solution you can see my attempt in the second picture uploaded. i don't think i even got it right
  24. anemone

    MHB Can you factorize this trigonometric expression?

    Factorize $\cos^2 x+\cos^2 2x+\cos^2 3x+\cos 2x+\cos 4x+\cos 6x$.
  25. Greg

    MHB Trigonometric sum with a product as the argument

    Prove $$\sum_{n=0}^N\cos(nx)=\csc\left(\dfrac x2\right)\sin\left(\dfrac{(N+1)x}{2}\right)\cos\left(\dfrac{Nx}{2}\right)$$ I've tried working from the RHS with various identities but haven't managed to come up with anything that works. I suspect this problem involves some trigonometry that I...
  26. Eclair_de_XII

    Can you derive a trigonometric function from its inverse dx?

    Homework Statement Arbitrary derivative of inverse trigonometric function: (sin-1x) = 1/(√1 - x2) Homework Equations f-1(f(x)) = 1/f`(x) The Attempt at a Solution So basically I learned about derivatives of trigonometric functions in class, and I thought maybe this would work: deriving the...
  27. G

    MHB Inverse trigonometric functions

    What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct? The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$ $\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$
  28. H

    Integration of trigonometric function

    Homework Statement I have included the LaTex version of the problem. \int \frac{sin^2 x}{1+cos^2 x} dx Homework Equations Simplifying fraction Partial fractions The Attempt at a Solution I have uploaded my attempt at the solution.
  29. anemone

    MHB Can You Prove $\sin 1+\sin 2+\sin 3+\cdots+\sin n<2$?

    Prove that $\sin 1+\sin 2+\sin 3+\cdots+\sin n<2$.
  30. kaliprasad

    MHB Prove: $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$

    Prove that $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$
  31. Nono713

    MHB Divergence of a trigonometric series

    Show that this series diverges: $$\sum_{n = 0}^\infty \cos \left ( n^2 \right )$$ (in the sense that it takes arbitrarily large values as $n \to \infty$)
  32. Matejxx1

    Trigonometric equations (finding angles)

    Homework Statement ok so my professor gave me this problem to solve, it goes like this :(I will also have a picture below) In the square (ABCD) is a point P which divides the side BC into 2 halves and point R which divides the side CD into 2 halves The angles at APB and ARB are the same...
  33. A

    Understanding Trigonometric Substitution

    When using trigonometric substitution in calculus you're supposed to always keep in mind the domain of the angle. In the case of √(x2-a2) (where "a" is a number >0) you use x=a⋅arcsec Θ for the substitution. For trigonometric substitution, textbooks state that the domain of Θ must be...
  34. Taryn1

    MHB Write a trigonometric expression as an algebraic expression

    This problem probably should be easy, but I don't remember learning the basic way to do these problems: Write the trigonometric expression as an algebraic expression: cos(arccos x + arcsin x) The answer is zero, but I don't know how to get there...
  35. anemone

    MHB Is This Trigonometric Identity Valid for All Values?

    Let $\dfrac{\cos^4 a}{x}+\dfrac{\sin^4 a}{y}=\dfrac{1}{x+y}$ for all real $a,\,b,\,x,\,y$. Prove that $\dfrac{\cos^8 a}{x^3}+\dfrac{\sin^8 a}{y^3}=\dfrac{1}{(x+y)^3}$
  36. Sollicitans

    Linear Independence of trigonometric functions

    Homework Statement There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x. Homework Equations...
  37. D

    Trigonometric Problem: Solving for sin(x/2)*cos(5p/4) with Given Conditions

    Homework Statement cos(x - 3p/2) = - 4/5 p <x< p/2 sin(x/2)*cos(5p/4)= ? Homework Equations The Attempt at a Solution I made it as far as to determine that sinx= 4/5 and cosx = - 3/5 but can't seems to progress any further. I am looking for an easier way to find the solution without having to...
  38. MironeDagains

    Solve Trig Equation with 2 & -Π/6 Inside Brackets

    http://www5a.wolframalpha.com/Calculate/MSP/MSP238521i5b83i951f19c3000010ca05be63f0bfc0?MSPStoreType=image/gif&s=10&w=219.&h=85. How do I solve this? I know the answers, as Wolphram Alpha has given me only the answers without any steps to how they derived those answers. I know that sin(x)=√3/2...
  39. Oribe Yasuna

    Integrating dx / (4+x^2)^2 using Trigonometric Substitution

    Homework Statement Evaluate the integral: integral of dx / (4+x^2)^2 Homework Equations x = a tan x theta a^2 + x^2 = a^2 sec^2 theta The Attempt at a Solution x = 2 tan theta dx = 2sec^2 theta tan theta = x/2 integral of dx / (4+x^2)^2 = 1/8 integral (sec^2 theta / sec^4 theta) d theta =...
  40. S

    Solving hyperbolic trigonometric equations

    Homework Statement Show that the real solution ##x## of $$tanhx=cosechx$$ can be written in the form ##x=ln(a \pm \sqrt{a})## and find an explicit value for ##a##. Homework Equations $$cosh^{2}x-sinh^{2}x=1$$ $$coshx=\frac{e^{x}+e^{-x}}{2}$$ The Attempt at a Solution I reduced the original...
  41. F

    Limit with trigonometric and polynomial function.

    Homework Statement For $$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 2 }+{ e }^{ -{ x }^{ 2 }\sin ^{ 2 }{ x } } }{ \sqrt { { x }^{ 4 }+1 } } } $$, determine whether it exists. If it does, find its value. if it doesn't, explain. Homework Equations Sand witch theorem and arithmetic rule...
  42. A

    Inverse trigonometric function integration

    I'm struggling to solve the following integral ∫ x/(√27-6x-x2) my attempt is as follows: ∫x/(√36 - (x+3)2) = ∫1/ √(36 - (x+3)2) + ∫x+1/ √(36 - (x+3)2) = arcsin (x + 3)/6 + this is where I got stuck.
  43. C

    Integration via Trigonometric Substitution

    Homework Statement Evaluate \int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx via trigonometric substitution. You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution. Homework EquationsThe Attempt at a Solution Letting x=sinθ...
  44. anemone

    MHB Can You Solve This Trigonometric Equation for $x$?

    Solve for $x$ such that $2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$ for $0\lt x \lt 180^\circ$.
  45. C

    MHB Trigonometric Identities Problem

    1) If \tan(\pi/4)=1, find \cot(\pi-\pi/4). 2) If \cot(17^{\circ}) = 3.2709, find \tan(73^{\circ}) 3) If \cot(\theta) = \frac{-9}{2} with \theta in Quadrant II, find \sin (\theta) --------------------------------------------- I really have no idea how to solve any of these problems. I have...
  46. O

    Derivative of a trigonometric function

    Homework Statement \frac{d}{dx}7.5sin(\frac{pi}{10}x) The Attempt at a Solution 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) Maximum: f'(x) = 0 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) = 0 7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x ** (\frac{pi}{10}\frac{10}{pi})7.5(90) = x (1)(7.5)(90) = x...
  47. M

    Integration: inverse trigonometric functions

    Homework Statement ∫(t/√(1-t4))dt Homework Equations ∫(du/√(a2 - u2)) = arcsin (u/a) + C ∫(du/(a2 + u2) = (1/a) arctan (u/a) ∫(du/(u√(u2 - a2))) = (1/a) arcsec (|u|/a) The Attempt at a Solution Edit: I meant to write u where t2 is[/B]
  48. Rectifier

    Proving Trigonometric Identity: tan(x/2) = (1-cos(x))/sin(x)

    The problem Show that the left side is equal to right side ## tan (\frac{x}{2}) = \frac{1-cos(x)}{sin(x)} ## The attempt ##\tan(\frac{x}{2}) = \frac{ sin(\frac{x}{2}) }{ cos (\frac{x}{2}) } = \frac{ sin^2(\frac{x}{2}) }{ cos ^2 (\frac{x}{2}) } = \frac{\frac{1-cos(x)}{2}}{\frac{1+cos(x)}{2}} =...
  49. P

    Why do we assume certain values for theta and x in trigonometric substitutions?

    http://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx In example one, the author drops the absolute value bars and makes the following statement: "Without limits we won’t be able to determine if ##\tan{\theta}## is positive or negative, however, we will need to eliminate them in...
  50. C

    MHB Simple Trigonometric Identities

    If (sinΘ) = 2/3 with Θ in quadrant 1, find (secΘ)[/SIZE] Θ = theta Completely new at trigonometric identities, would be a great help!
Back
Top