What is Trigonometric functions: Definition and 163 Discussions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

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  1. A

    How can I know when to use different trigonometric functions?

    Can someone please explain to me why in this problem, for example, the sine and cosine functions are used in equating the force's components? I am having a hard time solving for these unknown forces because of my rusty trig skills. It asks to "determine the maximum weight of the flowerpot...
  2. V

    Understanding Trigonometric Functions and Their Geometric Meaning

    I'm not sure if this is the correct section for this thread since this isn't homework, but my question is very basic, so I think this section is suitable. I have two questions regarding the trigonometric functions (sinx,cosx,tanx etc). 1) What is the geometric meaning (i.e in the context...
  3. anemone

    MHB Solving exponential (of trigonometric functions) equation

    Hi MHB, Solve $(2+ \sqrt{2})^{(\sin x)^2}-(2- \sqrt{2})^{(\cos x)^2}=\left( 1+ \dfrac{1}{\sqrt{2}} \right)^{\cos 2x} -(2-\sqrt{2})^{\cos 2x}$. This problem vexes me much because the only way that I could think of to solve this problem would be by substituting $(\sin x)^2=u$, and from there, I...
  4. J

    Parity of inverse trigonometric functions

    When I place the trigonometric functions in the "wolfram google", it informs the parity of the function, so, sin(x), sinh(x) -> odd cos(x), cosh(x) -> even tan(x), tanh(x) -> odd cot(x), coth(x) -> odd sec(x), sech(x) -> even csc(x), csch(x) -> odd arcsin(x), arcsinh(x) -> odd...
  5. talknerdy2me

    Rearranging Trigonometric Functions

    This is my very first post - so i hope I don't break any rules - its more of a formula rearranging question/confirmation so here goes... Homework Statement Currently working on friction - static/kinetic - so in my textbook it states in a side bar "info bit" that tanθ=sinθ/cosθ my...
  6. K

    Definite integration of Trigonometric Functions

    Homework Statement [0,1]∫(3x)dx/(4-3x)^1/2 (3xdx divided by square root of 4-3x) Homework Equations The Attempt at a Solution I could not get the bookish answer of that...actually my answer was wholly different... i let 4-3x (without square root) = t and then use substitution...
  7. J

    Derivative and trigonometric functions

    Hellow! If we can equal the first derivative with a trigonometric function: \frac{dy}{dx}=tan(\theta) So, the second derivative is equal to which trigonometric function? \frac{d^2y}{dx^2}=? Thanks!
  8. K

    Integration problem in trigonometric functions

    Homework Statement please help me with this integration problem? ∫(1/sinx+ cosx) dx Homework Equations i don't know any proper substitution in this question,maybe there are none The Attempt at a Solution i tried rationalizing and it has got me this far...
  9. B

    MHB Integration involving trigonometric functions

    any hints on how to work out this problems. $\displaystyle\int\frac{dx}{(1-sinx)^2}$ $\displaystyle\int\sin x\sin2x\sin3x dx$ thanks!
  10. T

    Derivative of Trigonometric Functions

    Homework Statement d/dx(sec(x)/1+tan(x) Evaluate at x=∏/6 Homework Equations The Attempt at a Solution ((1/cos(x))(1+tan(x))-(sec(x))(sin(x)/cos(x)))/(1+tan(x))^2 ((1/cos(x))-(sec(x))(sin(x)/cos(x)))/(1+tan(x)) I used the quotient rule and reduced what I could. Have I done this...
  11. paulmdrdo1

    MHB How to Calculate the Values of Other Trigonometric Functions Using Given Values?

    find the values of other five trig functions $\csc\theta=-2\,and\, \cot\theta>0$ my solution $x=-2$ $y=-1$ $r=\sqrt{5}$ $\displaystyle \sin\theta=-\frac{1}{\sqrt{5}}$ $\displaystyle\cos\theta=-\frac{2}{\sqrt{5}}$ $\displaystyle\cot\theta=2$ $\displaystyle\tan\theta=\frac{1}{2}$...
  12. K

    Areas Bounded by Trigonometric Functions.

    I will do my best to describe the problem I am working on. The problem is not from a textbook or anything but something I am working on independently to strengthen my first year calculus knowledge. What I did is I took sin(x) and -sin(x) and graphed them together. Sin(x) and -sin(x)...
  13. F

    Dividing by trigonometric functions

    Hello. I was doing a (simple) physics problem and stumbled with a mathematical problem. I was doing a projectile motions problem and I have set up my equation like this:Δx = Vi (cosθ) (t) 270= 25cosθ t t = 270 / (250cosθ) And this is where I'm having problems. I know from my high school trig...
  14. S

    MHB Applying Trigonometric Functions

    Okay, so I know how to apply basic trigonometric functions to solve right triangles. However, I am not quite sure how to manipulate trigonometric functions to solve for more complex questions involving non-right triangles, like the question I have attached. I have to use the basic trigonometric...
  15. V

    Trigonometric functions ->sine function EASY

    Trigonometric functions -->sine function EASY! Homework Statement Consider the trigonometric function f(t)=-1+4sin(0.5π(t-1)). (a) What is the amplitude of f(t)? (b) What is the period of f(t)? (c) What are the maximum and minimum values attained by f(t)? Homework Equations...
  16. Petrus

    MHB Tangent lines of trigonometric functions

    Hello, I got problem with A homework "find an equation of the tangent line to curve at the given point. $y=sec(x)$. $(pi/3,2)$ progress: $y'=sec(x)tan(x)$. So basicly that sec(x) don't say me much so i rewrite it as $1/cos(x)$ $y'=1/cos(x)•tan(x)$ now i can put $pi/3$ on the function to...
  17. D

    Derivative of trigonometric functions

    Homework Statement g(x) = 4∏ [cos(3∏x) sin (3∏x)]The Attempt at a Solution g(x) = 4∏ [cos(3∏x) sin (3∏x)]' 4∏{[cos (3∏x)][sin(3∏x)]' + [sin(3∏x)][cos(3∏x)]'} = 4∏{[cos (3∏x)][cos(3∏x) . (3∏)] + [sin(3∏x)][-sin(3∏x) . (3∏)] = Now, my question is: Can I combine the numbers and have the...
  18. Petrus

    MHB Trigonometric functions: Sec, Cot, Csc

    Hello, Im currently on chapter about derivate trigonometric functions. It have been hard for me to understand this sec,cot,-csc? Why do you rewrite example $1/cos^2x$ as $sec^2x$? when I get like sec,csc etc i kinda feel i have no clue what it means. Then you think what do Petrus mean? example I...
  19. S

    Help with derivative of trigonometric functions

    Find y^{I} (sinxsecx)/1+xtanx The supplied answer is 1/(1+xtanx)^{2} I got stuck with an extra x on top at the end. Where did I mess up at? y^{I}(sinxsecx)/1+xtanx = [1+xtanx*f^{I}(sinxsecx)-sinxsecx*f^{I}(1+xtanx)]/(1+xtanx)^{2} =...
  20. MarkFL

    MHB Rebekah's question at Yahoo Answers involving inverse trigonometric functions

    Here is the question: Here is a link to the question: Given 0<x<orequalto 1, determine the value of inversine (x) + inversetan (squareroot(1-x^2)/x)? - Yahoo! Answers I have posted a link there to this topic so the OP may find my response.
  21. E

    Capitalising trigonometric functions

    I have always capitalised the first letter of my trigonometric functions, for example, writing Sinθ as opposed to the usual sinθ. Is it wrong to capitalise them? Does it make a difference in meaning?
  22. K

    How to write powers of inverse trigonometric functions?

    Does ##(\sin^{-1}\theta)^2 =\sin^{-2}\theta## ?
  23. M

    Signs of the trigonometric functions of the angle -200

    Determine the signs of the trigonometric functions of the angle -200° in standard position. sin(-200°) cos(-200°) tan(-200°) csc(-200°) sec(-200°) cot(-200°) Information: Q1= (All Positive) Q2= (sin csc) Q3= (tan cot) Q4= (cos sec) Answer: sin(-200°) = Positive cos(-200°) =...
  24. M

    Values of the six Trigonometric Functions

    Find the values of the six trigonometric functions of an angle θ in standard position whose terminal side is containing the points (-3,0) Sinθ= Cosθ= Tanθ= Cscθ= Secθ= Cotθ= I believe the following are correct But I am not sure, Please give insight Sinθ= 0/3 Cosθ= -3/3 Tanθ= 0/3...
  25. B

    Limits Involving Trigonometric Functions (identities)

    Does anyone know of websites where I can find many problems on the topic in the title line (my textbook has far too few)? Thanks!
  26. A

    Trigonometry Word Problem involving Inverse trigonometric functions

    Homework Statement A ship is d feet from a dock (horizontal distance). The dock is 40 feet above sea level. The angle of depression from the dock to the ship is θ. Write θ as a function of d. Homework Equations This question is accompanied with choices. The choices are (A) θ = 40arctan(d) (B)...
  27. M

    Dif.eq. with trigonometric functions involving y

    I tried to solve it but confused. Pls. help me to solve this equation: dy/dx + (e^x)*Sec(y) = Tan(x); (hint: integrating factor is e^-ax, and a is unknown, a ε ℝ, find it, solve the equation) Thnx.
  28. T

    Differential Equation Involving Trigonometric Functions

    Homework Statement Solve the differential equation: \frac{dy}{dx} = cos^2 (x) cos^2 (2y) The Attempt at a Solution I rewrote the equation \frac{dy}{cos^2 (2y)} = sec^2 (2y) = cos^2 (x) dx. Then I integrated, \frac{tan(2y)}{2} = \frac{1}{2} (x + sin(x)cos(x)) + c. Then I solved for y...
  29. I

    Few Trigonometric Functions that I can’t solve involving identities? helpp

    1. Sin^2(x) = 3 – x Answer: 2.97 Attempts: 1-cos^2(x) = 3 – x cos^2(x) - x + 2 = 0 Factored it and got x = pi = 3.14 It’s a multiple choice question, and other answers were 3.02,3.09 which are few decimal places off so the answer must not be pi since it's not even a choice. Is the...
  30. B

    Derivative involving inverse trigonometric functions

    Homework Statement Find the derivative of: sqrt(x^2-4)-2tan^-1{.5*sqrt(x^2-4)} Homework Equations U'/1+U^2 U'=x/2sqrt(x^2-4) 1+U^2=x^2 The Attempt at a Solution I combined the above components but my answer is incorrect. I feel that I might have the wrong answer for...
  31. J

    Derivation of a formula with trigonometric functions

    Hi everyone, Homework Statement My problem is just to derive a simple formula, which is http://www.texify.com/img/%5Cnormalsize%5C%21%28-1%29%5E%7Br%28r%2B1%29/2%7D%20%3D%5Csqrt%7B2%7D%20%5Cmbox%7Bcos%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%282r%2B1%29.gif Here r is a positive integer. The Attempt...
  32. mnb96

    System of two differential equations with trigonometric functions

    Hello, do you have any strategy to suggest in order to solve the following system of partial differential equations in x(s,t) and y(s,t)? \frac{\partial x}{\partial t} = x - \frac{1}{2}\sin(2x) \frac{\partial y}{\partial t} = y \; \sin^2(x) (note that the partial differentiation is always with...
  33. Blandongstein

    Integral involving trigonometric functions.

    I found this question on a website. Homework Statement Prove that \displaystyle \int_{0}^{\pi}\frac{1-\cos(nx)}{1-\cos(x)} dx=n\pi \ \ , n \in \mathbb{N} 2. The attempt at a solution Here's my attempt using induction: Let P(n) be the statement given by...
  34. D

    Integration of inverse trigonometric functions

    Homework Statement ∫ (x+2)dx/√(4x-x2) Homework Equations why was the -2 in -2(x-2) was ignored? The Attempt at a Solution so first i let u= 4x-x2 then, du=4-2x = -2(x-2) so to get (x+2) i equate it to (x-2)+4 so ... ∫ (x+2)dx/√(4x-x2) = ∫(x-2)+4dx/√(4x-x2) = ∫...
  35. P

    Question about integration with inverse trigonometric functions

    I'm self-studying Calculus and would like to ask some doubts about the following question: Homework Statement If, in t seconds, s is the oriented distance of the particle from the origin and v is the velocity of the particle, then a differential equation for harmonic simple motion is...
  36. M

    Trigonometry Question (Trigonometric Functions)

    Homework Statement Hello again, not looking for anyone to solve this problem this time I just need to know what the operation I'm supposed to do is called, so I can go research it and learn it. If sin(t) = -12/13 with 3pi/2 < t < 2pi, find the following: b) cos(t - 11pi/6) Homework...
  37. D

    Limits question involving trigonometric functions

    f(x)= ((cosx)^{2}+1)/e^{x}^{2} So for the limit of f(x) as x→∞ I would just input ∞ for x. I'm confused after this though, wouldn't it just be ∞/∞ = 1? the next part says show that there exists a number c ε (0,1) that f(c)=1 I don't know what this is asking for me to solve.
  38. N

    Differentiating Trigonometric Functions: Find the Derivatives

    find the derivatives of differentiation of trigonometric functions 1. y=cos(3x^2+8x-2) 2. y=tan^3 2x 3. y=sin5x sin^5 x 4. y=Square root of 4sin^2x+9cos^2x help here please.. i can't understand trigonometric functions sorry admin or moderator, i just search the net on how...
  39. X

    Trigonometric functions using accel & time

    Homework Statement A skier races down an 18 degree ski slope. During a 5.0s interval, the skier accelerates at 2.5m/s squared. What are the horizontal and vertical components of the skiers acceleration during this time?Homework Equations d = ViT + 1/2aT^2 d=distance Vi=initial velocity...
  40. M

    Can I Just Use Trigonometric Functions Without Understanding Them?

    Hi, For my exams, I am provided with a list of trigonometric functions. I do know at least a good half of those I'm supposed to know but I was wondering if I could get away with *just* knowing how to use them? I know things like sin^2(x)=1 - cos^2(x) or sin(A+B) = sinAcosB + cosAsinB but...
  41. 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...
  42. 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...
  43. 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...
  44. 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)}...
  45. 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...
  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. 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...
  48. I

    Limits involving trigonometric functions

    Homework Statement 1) lim sin^2(3x)/x^2 x->0 2) lim 2sin5x/(3x-2tan2x) x->0 Homework Equations lim sinx/x = 1 x->0lim tanx/x = 1 x->0 The Attempt at a Solution We just began working with limits, and we haven't covered much of trig functions at all, but our prof gave us...
  49. Y

    Expressing cos(2π/n) without using trigonometric functions

    I've read a bit about constructibe polygons, and that a regular n-gon can be constructed with compass and straightedge if and only if trigonomatric functions of 2π/n can be expressed with square roots and basic arithmatic alone. That is possible if and only if n is the product of distinct Fermat...
  50. K

    Intergration of higher powers of trigonometric functions

    Hello, What's the easiest way to evaluate an integral like this? \int_{\frac{-\pi}{2}}^{0}\cos^{10}x dx The only method I can think of is to expand the \cos^{10} x using trigonometric identities, and getting \frac{1}{32}\left(1-\cos2x\right)^5. I tried subbing u=1-\cos2x but I doesn't seem to...
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