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. B

    I Finding the inverse tangent of a complex number

    Let z=x+iy, and w=u+iv. I am looking for a formula to find the arctangent of z, or w=arctan(z). I want the results of u and v to be in terms of trigonometric and hyperbolic functions (and their inverses) and not in terms of logarithms. The values u and v should be functions of x and y.
  2. WMDhamnekar

    Evaluation of integral having trigonometric functions

    R is the triangle which area is enclosed by the line x=2, y=0 and y=x. Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v## ## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
  3. P

    I Finding a bound for a fraction

    Hi, I have given the following, which I would like to show that this estimation is correct, where ##|\theta| \leq \frac{\pi}{^2}## and ##M \geq 1##: $$\frac{1}{M^2}\frac{\sin^2(M\theta)}{\sin^2(\theta)} \geq \frac{4}{\pi^2}$$ I would approach an estimation of the denominator via ##\sin(x)...
  4. K

    MHB Proving trigonometric functions

    How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?
  5. John Greger

    I Units of trigonometric functions?

    What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?
  6. M

    Inverse trigonometric functions

    Create one equation of a reciprocal trigonometric function that has the following: Domain: ##x\neq \frac{5\pi}{6}+\frac{\pi}{3}n## Range: ##y\le1## or ##y\ge9## I think the solution has to be in the form of ##y=4sec( )+5## OR ##y=4csc( )+5##, but I am not sure on what to include...
  7. xyz_1965

    MHB Is tan(x)^2 proper notation for the trig function tangent squared?

    Is tan^2 (x) the same as tan(x)^2? Note: I could have used any trig function. I know that tan^2 (x) means (tan x)^2. What does tan (x)^2 mean? Is it proper notation?
  8. agnimusayoti

    Absolute value of trigonometric functions of a complex number

    So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then 1. Real part: ##\sin x \cosh y## 2. Imaginary part: ##\cos x \sinh y## If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }## How to...
  9. Avaro667

    A Elliptic trigonometric functions as basis for function expansion ?

    Hey everyone . So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking ! So i was...
  10. 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...
  11. 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...
  12. M

    B Understanding the Odd and Even Nature of sin(x^3)

    Hello, would you please explain how to determine if sin ##x^3## is odd or even? Is there anyway to understand it without drawing the graph? Thank you.
  13. 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!
  14. navneet9431

    Problem in finding a general solution

    Homework Statement Homework Equations General Formula for Tan(a)=Tan(b) The Attempt at a Solution See the question I have uploaded. I have tried solving it this way, Firstly I applied the Quadratic Formula to get, Now we have two cases, CASE-1 When So General Formula here will...
  15. 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...
  16. 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...
  17. shihab-kol

    B Why Does tan x Have This Domain?

    I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I} The graph however shows that for every value of x , the function takes on a value .So, why is the domain like this?
  18. Const@ntine

    Transverse Wave: Time difference between two points

    Homework Statement A transverse wave that is propagated through a wire, is described through this function: y(x,t) = 0.350sin(1.25x + 99.6t) SI Consider the point of the wire that is found at x= 0: a) What's the time difference between the two first arrivals of x = 0 at the height y =...
  19. H

    A circle is circumscribed around triangle ABC, find length?

    some formula related I tried to draw the problem can anyone give me clue how to solve it?
  20. Wrichik Basu

    A problem in Inverse Circular Functions in Trigonometry

    Homework Statement :[/B] Solve for ##x ##: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$ Answer given: ##0## or ##\frac {1}{2}##. Homework Equations :[/B] All relevant formulae on inverse circular functions may be used. The Attempt at a Solution :[/B] Please see the pic below...
  21. Vital

    Show that the function is a sinusoid by rewriting it

    Homework Statement Hello! I am doing exercises on sinusoid functions from the beginning of Trigonometry. I hoped I understood the topic, but it seems not quite, because I don't get the results authors show as examples for one of possible answers, as there can be a few answers to the same...
  22. Vital

    Graphs of sin and cos, how to set points for x values

    Homework Statement Hello! I am at the topic on graphing trigonometric functions. Exercises are rather easy at this point, but I have a problem deciphering how authors of the book choose points for x values. Please, take a look at few examples (including screen shots I attach), and, please...
  23. G

    I Trigonometric series with normalised coefficients

    Hi all, I have a trigonometric function series $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$ with the normalization condition $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$ and ##\Lambda_l## being monotonic decrescent...
  24. A

    Trigonometric functions and integrals

    Homework Statement I'm searching for the integral that gives arcosu Homework Equations as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu derivative of arccosu = -u'/[1-u^2]^0.5 + C derivative of arcsinu= u'/[1-u^2]^0.5 The Attempt at a Solution when I type the -u'/[1-u^2]^0.5 on the online integral...
  25. pairofstrings

    Visualize this type of Combined Trigonometric Functions

    Homework Statement Show that sin 600° . cos 330° + cos 120° . sin 150° = - 1 Homework Equations I know that sinΘ = opposite/hypotenuse and cosΘ = adjacent/hypotenuse. The Attempt at a Solution I am equipped with knowledge about what sinΘ and cosΘ is from right angled triangle. I stand in...
  26. ItsAnshumaan

    Graph of trigonometric functions

    This is not a homework question but a general doubt. Suppose we have a function y = pcosx, where 'p' is an arbitrary constant. So my question is how will the graph of this function change with different values of 'p'? This doubt can also be extended for other functions like y = pex, y = p...
  27. anemone

    MHB Evaluate a floor function involving trigonometric functions

    Evaluate \left\lfloor{\tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right)}\right\rfloor. Hi MHB, I don't know how to solve the above problem, as I have exhausted all possible methods that I could think of, and I firmly believe there got to be...
  28. N

    I How Do You Solve the Integral of 1/(y+cos(x))^2?

    First part of the question was to work out the integral 1/(y+cos(x)) between x=0 and x=pi/2 by using the substitution t=tan(x/2). Got this to be \frac{2}{\sqrt{y^2-1}}arctan(\sqrt{\frac{y-1}{y+1}}) The next question says HENCE find integral with the same limits of \frac{1}{(y+cos(x))^2} Ive...
  29. Alanay

    How do I calculate inverse trig functions?

    On the paper I'm reading the arctan of 35 over 65 is approx. 28.30degrees. When I use the Google calculator "arctan(35/65)" gives me 0.493941369 rad. What am I doing wrong?
  30. Dean Whaley

    Solve for a constant in an equation

    Im trying to solve for a constant in an equation and it involves taking the arctanh(6.55) and my calculator is giving me an error, is there a way around this?
  31. 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 $\frac{2\pi}{3}$
  32. 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...
  33. A

    What is causing confusion in solving for projectile motion?

    Hello, I am a first year science teacher doing my best with teaching physics for the first time (my degree is chemistry but I am in a very small school). I am teaching projectile motion. I was creating a worksheet and trying to solve a problem I made up when I realized something wasn't working...
  34. L

    How Does cos(θ)sin(θ+φ)-sinθcos(θ+φ) Simplify to sinφ?

    Can someone please show me, step-by-step, how cos(θ)sin(θ+φ)-sinθcos(θ+φ) simplifies to sinφ? I know I have to use trig identities, and I got to cos2θsinφ-sin2θcosφ but I'm not sure where to go from there. Thanks
  35. M

    Is there an error in my textbook?

    Homework Statement Homework Equations Doesn't the integral of sec x tan x equal sec x? The Attempt at a Solution
  36. 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]
  37. P

    Inverse trigonometric functions

    I am familiar with the importance of the following inverse circular/hyperbolic functions: ##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##. However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on. Given any equation of the form...
  38. W

    Trigonometric functions (identity&equations)

    1) Problem: given that x is an obtuse angle for which cos^2x/(1 + 5sin^2x) = 8/35, find the value of cosx/(1 - 5 sin x) without evaluating x. 2) Relevent equations: sin(-x) = - sin x cos(-x) = cos x sin(180° - x) = sin x cos(180° - x) = - cos x sin^2x + cos^2x = 1 3) Attempt: cos^2x/(1 +...
  39. J

    Can Euler-Lagrange Equations Explain Mirages?

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
  40. L

    MHB Trigonometric functions problem

    Hello, I am trying to solve this. This material is not covered in my class, but I still want to know how to do it. If cos(t)=$\frac{-9}{10}$ where $\pi$ <t<$\frac{3\pi}{2}$ find the values of cos(2t)= sin(2t)= cos($\frac{t}{2}$)= sin($\frac{t}{2}$)= Give exact answers, do not use decimal...
  41. Adriane

    I'm having trouble creating my own trig function

    I'm developing my own trigonometric function concerning a "real world" problem of my choosing. I decided to go with the orbit of Neptune around the sun. I just don't know how to develop the equation itself, like if it would be sine or cosine? I'm just lost as to where to begin. If anyone can...
  42. Math Amateur

    MHB HIGHLY Rigorous Treatment of the Trigonometric Functions

    I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ... Can someone please suggest (i) an online...
  43. T

    MHB Adding Trigonometric Functions

    I've muddled my way through the majority of my weekend assignment and I'm stuck on a problem where I need to add two formulas together. 1.) 20-10cos(x*pi/4) 2.) 30+20sin(x*pi/4) I end up with a sinusoidal function which I can then graph and determine the max, min, etc. We recently went over...
  44. T

    MHB Finding expressions for the five other trigonometric functions....

    So here's the question: Suppose cos(θ) =x/4. Find expressions for the other five trigonometric functions in terms of x. In our practice problems we never had a variable x used and we were able to use the pythagorean theorem to determine the final side of the triangle and simply figure out the...
  45. N

    Understanding the change from cot graph to tan graph

    Homework Statement Suppose the function is y = a cot k(x−b) Then (give exact answers; you can type pi for π): a = b = k = Suppose the function is y = a tan k(x−b), where b > 0. Then: a = b = k = The Attempt at a Solution Then (give exact answers; you can type pi for π): a = 4...
  46. A

    Can anybody check this proof for a Sine limit?

    Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...
  47. chwala

    Proving trigonometric functions

    Mod note: Moved from a technical math forum, so this post is missing the homework template i am trying to prove that ##1/sec∅-tan∅ ≡ sec ∅ + tan∅## this is how i attempted it, i tried to show that the left hand side is equal to the right... ## 1/ 1/(cos∅-sin∅)/cos∅## where i end up with ##...
  48. T

    Taylor Polynomial of 3rd order in 0 to f(x) = sin(arctan (x))

    The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...
  49. Mr Davis 97

    How do we define trigonometric functions?

    I'm having a problem understanding exactly why trig functions are defined the way they are. Of course, the definition in terms of 0 to 90 degree angles within right triangles is easy: the functions just give the ratio of the sides given the angle. However, I don't understand how or why trig...