Talk:Trigonometric functions

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Former featured article Trigonometric functions is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.
Main Page trophy This article appeared on Wikipedia's Main Page as Today's featured article on March 6, 2004.
Article milestones
Date Process Result
December 12, 2003 Featured article candidate Promoted
September 20, 2004 Featured article review Kept
July 19, 2008 Featured article review Demoted
Current status: Former featured article
WikiProject Mathematics (Rated B-class, Top-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
Top Importance
 Field:  Analysis
One of the 500 most frequently viewed mathematics articles.
A selected article on the Mathematics Portal.
Wikipedia Version 1.0 Editorial Team / v0.5 (Rated Start-class)
WikiProject icon This article has been reviewed by the Version 1.0 Editorial Team.
Start-Class article Start  Quality: Start-Class
Checklist icon
 ???  Importance: not yet rated

Placement of sin 18°[edit]

Should sin 18° not be placed in the list alongside 15°, 54° and 75°, because it is as complex as sin 54°? — Preceding unsigned comment added by (talk) 02:22, 4 May 2012 (UTC)

They are in numerical order in the individual tables, not according to complexity. They were rearranged into numeric order before I read this. Dmcq (talk) 10:44, 4 May 2012 (UTC)

Better unit circle definition[edit]

Isn't the whole beauty of the unit circle definition that you can avoid mentioning angles altogether? This is the approach used in Lipman Bers' Calculus textbook (volume one):

1. Define the unit circle.

2. The length of the unit circle is 2pi.

3. Define P(\theta) to be a point on the circle of length \theta, obtained by moving counter-clockwise from (1,0)

4. Define sin and cos as follows for point P(\theta) = (x,y): sin(\theta) = y cos(\theta) = x

5. Derive properties of sin and cos from the geometry of the unit circle (e.g. sin^2+cos^2 = 1 because x and y are on the unit circle)

6. Expand the definition to right triangles by defining a coordinate system in which the length of the hypotenuse is 1 (the radius of the unit circle) and the origin is...

7. Expand sin and cos to all triangles

So you've described theta as a distance along the unit circle instead of an angle. So why is that such a wonderful idea? Dmcq (talk) 17:16, 9 January 2012 (UTC)
Note that the definition of angle is arc length scaled to a unit circle, so this isn't even really a change, it is just avoiding defining a word. — Steven G. Johnson (talk) 17:25, 9 January 2012 (UTC)

Replacing the image of the sawtooth wave[edit]

I've created a replacement for the image File:Sawtooth Fourier Analysis.JPG which is in the periodic functions section of this article. Here it is:

Sawtooth Fourier Animation.gif

I'll wait for a day or two before putting it in to make sure I'm not stepping on anyone's toes. Of course I'll modify the caption too because it is not quite right for the new animation.

4dhayman (talk) 00:07, 21 February 2012 (UTC)

Looks good to me. The caption is far too long anyway: anything over two or three lines should be made into article content, where it is easier to read and is better integrated into the article.--JohnBlackburnewordsdeeds 00:45, 21 February 2012 (UTC)
Okay, thanks, I'll go ahead and put it in then. No harm done if someone wants it reverted later. 4dhayman (talk) 01:07, 21 February 2012 (UTC)

tan 90°[edit]

I see in the tables that tan 90° is listed as infinity. While tan(x) approaches infinity as x approaches 90° degrees, tan(90°) is not defined as it involves sin(90°) / cos(90°) = 1/0 and anything divided by zero is not defined. (talk) 13:13, 9 May 2012 (UTC)

It says below the table that "projective infinity" is the intended meaning. Some mention of a pole of order 1 would also be helpful.--LutzL (talk) 14:36, 9 May 2012 (UTC)

I do accept the meaning, and the fact that certain published tables list it as infinity, but the fact that the article states cot(x) = 1/tan(x) and tan(x) = infinity, cot(x) would therefor be 1/inf (zero) and not infinity. These functions are undefined at an angle of 90°, due to divide-by-zero problems. (talk) 13:06, 10 May 2012 (UTC)

Connection to the inner product[edit]

In between Inverse functions and Properties and applications, I'd like to add a very short section titled Connection to the inner product:

In an inner product space, the angle between two non-zero vectors is defined to be

Any objections? -- UKoch (talk) 14:39, 17 October 2012 (UTC)

I have now added the section--as a subsection of Inverse functions, since it refers to the arccos function. -- UKoch (talk) 18:28, 5 December 2012 (UTC)

τ vs. π[edit]

A recent edit (diff) has changed the identities in the table at Trigonometric functions#Right-angled triangle definitions to use τ rather than π. When I reverted the first edit with that change, my summary was "τ is good but Wikipedia follows the mainstream and does not try to show a better way". Is there something I'm missing to justify using τ here? I'm aware that a number of people regard τ as much better (see pi#In popular culture), but is there a reason to use a symbol that would be a mystery to many readers here? Johnuniq (talk) 22:33, 12 February 2013 (UTC)

(Replies 2 years later) Hello. I'm sorry that I made that edit. I was watching Vi Hart and Numberphile videos, and I got really serious about т. I will always use π on wikipedia from now on. Max Buskirk (talk) 21:53, 16 February 2015 (UTC)


i wanted to learn something - completely impossible from this article, this is just a reference for those who know all of this material already. — Preceding unsigned comment added by (talk) 21:44, 10 March 2013 (UTC)

  • i agree, the definition is supposed to be comprehensible without too much reference or dependence on other "terms". it was obviously written by those who already understand the subject and can't intuit how to explain it for those who don't. (talk) 11:03, 15 May 2013 (UTC)

I would like to point out that an encyclopedia article is not supposed to be the first place to learn about something. First consult a textbook, then for things that a textbook might leave out, or might get wrong, or might be slanted about, then go consult the encyclopedia. Or, first consult the encyclopedia in order to get a very vague and general idea of what is involved in the topic, what it is about, and a list of textbooks or sources in its bibliography. So these comments are invalid. (talk) 05:06, 1 September 2014 (UTC)

Boy I hope that is not the purpose of Wikipedia. That would make it pretty useless. 4 July 2017 (JCBoone) — Preceding unsigned comment added by Joseph C Boone (talkcontribs) 21:50, 4 July 2017 (UTC)


It is possible to express the value of sin(1°) analytically. It can be obtained by solving the cubic equation, sin(3°) = 3sin(1°)-4sin3(1°). Therefore, trigonometric functions of all angles of integer degrees can be expressed analytically. --Roland 19:51, 10 June 2013 (UTC)

The request is for an explicit expression, not an implicit one as you propose. There is not explicit solution for this cubic equation. All rational fractions of pi resp. all angles of rational degree can be implicitely expressed as a solution of an algebraic equation.--LutzL (talk) 17:25, 11 June 2013 (UTC)
Yes, there is. Cubic equations can be solved analytically. Please refer to
A website actually gives the solution: --Roland 21:59, 11 June 2013 (UTC)
You are completely right, there is an analytical expression for sin(1°), which is sin(1°) (understood as the evaluation of the series). But the thread starter obviously wanted an algebraic expression, i.e., one only involving roots in addition to the usual arithmetic operations.--LutzL (talk) 18:07, 2 July 2013 (UTC)
Additionally, if you check the analytical formula for the root of the cubic equation, then you will see that you need an auxillary analytical number to express the solutions. Which is, ... , wait, ... , wait, ... , wait for it, ... , yes, exactly sin(1°).--LutzL (talk) 11:47, 3 July 2013 (UTC)
The linked article expresses, in a very complicated way, the trivial fact that cos(1°)+i*sin(1°) is one of the cubic roots of cos(3°)+i*sin(3°). However, there is no way to express this cubic root using only arithmetic operations and roots of positive real numbers.--LutzL (talk) 12:08, 3 July 2013 (UTC)

The implication in the article is that sin(1°) can be evaluated without computing trigonometric functions. As noted above, this is false; so it would be better to remove this paragraph as misleading. I have done so. cffk (talk) 16:39, 14 December 2015 (UTC)

No, this is true, sin(1°) can be evaluated without computing trigonometric functions. It can be evaluated by arithmetic operations, including square and cube roots (through Cardano formula. I have thus restored this paragraph and added some clarification. D.Lazard (talk) 17:16, 14 December 2015 (UTC)
Surely the evaluation of the cube root of a complex number involves trigonometry? For example the real part of the cube root of 3 + 4i is 5^(1/3)*cos(atan(3/4)/3). Can you write it in a form that doesn't involve trigonometric functions (without popping back into the complex plane)? cffk (talk) 18:18, 14 December 2015 (UTC)
The article Root-finding algorithm provides many (much more than ten) methods for computing the roots (including the complex ones) of a polynomial. All may apply to computing cube roots (equation ), and none involve any trigonometric function nor polar coordinates in the complex plane. This is related with the fundamental theorem of algebra, which has nothing to do with trigonometry. D.Lazard (talk) 21:08, 14 December 2015 (UTC)
But, these root-finding methods are all iterative ones! Surely the point of the paragraph in question was that sin(1°) can be computed in finite terms using the basic arithmetic operations and extracting real roots.
In addition, if this is an allowed method, why bother mentioning the Cardano formula? You could just solve the original cubic equation. Better yet, why not just solve the "simpler" polynomial x^180 + 1 = 0. cffk (talk) 21:31, 14 December 2015 (UTC)
I see that Trigonometric constants expressed in real radicals provides a useful summary: "All trigonometric numbers—sines or cosines of rational multiples of 360°—are algebraic numbers (solutions of polynomial equations with integer coefficients); but not all of these are expressible in terms of real radicals. When they are, they are expressible more specifically in terms of square roots." cffk (talk) 22:25, 14 December 2015 (UTC)
You are right about iterative methods. In fact, everything depends on what is chosen as basic arithmetic operations. Even square roots require iterative methods for being evaluated. The formulas of this section do not compute the values of trigonometric functions, but express them in terms of a finite number of symbols. in the formulas for angles multiple of 3° theses symbols are the integers, the symbols for addition, subtraction, multiplication and division, and square roots. As this is not sufficient for representing every number, people used to introduce other symbols (or other numbers), such as n-th roots (real or possibly complex), Bring radicals, algebraic numbers, and finally real and complex numbers. They were interested in characterizing the smallest class to which belongs the numbers that are widely considered. The relevant facts, here, are: sin(x) and cos(x) are expressible in terms of square roots if x is an integer multiple of 1°. They are expressible in terms of square roots and non-real cube roots if x is an integer multiple of 1°. They are expressible in terms of n-th roots (and therefore algebraic) if x is the product of 1° by a rational number (this is a non-elementary consequence of Galois theory). Otherwise, either x (measured in degrees) or both sin(x) and cos(x) are transcendental numbers (this is a corollary of Baker's theorem, 1967).
IMO, these results are more useful than the useless formulas for angles that are multiple of 3°. Maybe, I'll edit the article for inserting these results, when I'll get some time for that. D.Lazard (talk) 10:49, 15 December 2015 (UTC)
Your changes have improved the article, thanks. However, now the paragraph about solving a cubic to get sin(1°) is even more out of place. You can't list sin(1°) under "Explicit values" when its evaluation requires either an iterative method or trigonometry to compute a complex cube root. I recommend removing this paragraph again (but leave the earlier passage which says that sin(1°) is an algebraic number). Finally, please remember that there is a serviceable, if somewhat awkward, direct method for computing square roots by hand which generates the digits one at a time. cffk (talk) 15:33, 16 December 2015 (UTC)
"Value" and "evaluation" are ambiguous word to which, it seems that you give your own meaning. In fact, I understand that, for you, a value is a numerical approximation. Formally this is not true, as a value of sin(1°) must be equal to sin(1°), which is not the case of any approximation. On the other hand, it seems that you not consider as a value the expression for sin(3°) given in the article, while most mathematicians consider it as an admissible value. Nevertheless I have edited the article for replacing "value" by "expression" for avoiding any confusion. D.Lazard (talk) 17:05, 16 December 2015 (UTC)
My objection is that the expression for sin(1°) is implicit because, when writing it out as a real expression, it necessarily involves other trigonometric functions (see also the comments by LutzL above). A secondary objection is that this is an altogether useless result which just serves to clutter the article and to confuse the reader. cffk (talk) 18:32, 16 December 2015 (UTC)

I'm still dubious about the wisdom of listing sin 1° as having an explicit value. However, if the article is going to list any expression, let's, at least, make it as simple as possible. So I've replaced the messy business of solving a cubic equation with two simple expressions for sin 1°. This makes it plain that these definitions are circular. cffk (talk) 15:21, 19 December 2015 (UTC)

There is no definition in this section; so there is no circular definition. Moreover, one cannot say that a 90th root is something simple. On the other hand, the formula that you give is smart way to avoid using Cardano formula while getting a very similar result. It has also the advantage that it may generalized to a proof that the sine and cosine of any rational angle may be expressed in terms of nth roots. I'll edit this new version for clarifying that this is not circular and provide a (useless) explicit expression of sin 1° in terms of radicals. D.Lazard (talk) 23:03, 19 December 2015 (UTC)

Suggest adding figure numbers to all of the figures[edit]

Earlier today, I reverted a good-faith edit by who found the "a, b, h" labeling in the text to be totally obscure, so he/she subsituted "o" (meaning "opposite") for "a" and "a" (meaning "adjacent") for "b". This attempt by to clarify things actually worsened the confusion.

The real problem is that there is often little or no obvious connection between text and figures, which were drawn by different people and use different labeling conventions.

I suggest giving the illustrations figure numbers.

PRO: Numbering the figures will allow them to be unambiguously referred to in the text.
CON: I am not aware of any automatic numbering templates that will keep text numbering and figuring numbering in sync. If figures are added or deleted, a laborious search needs to be made through the text to make sure that all numbers match.

I believe that the PRO advantages outweigh the CONs. If a consensus agrees with me, I will add figure numbers.

Example articles that I have worked on that have required figure numbering include Interferometry, Michelson interferometer, Mach–Zehnder interferometer, Quadratic equation, Kaufmann–Bucherer–Neumann experiments, and so forth. A common feature of these articles has been the need to reference multiple figures from diverse points in the text. I believe that Trigonometric functions has the same requirement. Stigmatella aurantiaca (talk) 02:15, 2 July 2013 (UTC)

Is the massive animated figure appropriate?[edit]

There is a very large, very animated figure next to the section on the relationship to the exponential and complex numbers. This figure seems to me to be both (1) almost entirely original research and (2) using very unencyclopaedic language (ie "thru"). I don't think it makes a good addition to the article, not least because moving images very rarely are, and I think it should be removed, per policy on OR and appropriate language. Any disagreement? Quantum Burrito (talk) 22:32, 7 December 2013 (UTC)

trignometry is a life key — Preceding unsigned comment added by K.sarankathiravan (talkcontribs) 15:12, 23 January 2014 (UTC)

New animation, explaining sine and cosine as related to the unit circle, with their respective graphs[edit]

Cos, sin and the unit circle.

For what it's worth, I recently made this animation explaining cosine and sine in terms of the unit circle. Please, read the image's description on the image's page (just click the image) before making any remarks.

This is the only representation of both functions and their relation to the unit circle I could figure out that would:

1. Show the graph of both sin(θ) and cos(θ) in the usual orientation, where the horizontal axis represents θ and the vertical the value of the function.

2. The graphs shown, when animated, would not be drawn inverted when θ increases (the point in the unit circle moves counter-clockwise, as usual).

The "bent" way I used to represent cosine was necessary in order to have the graph y = cos(θ) in the usual orientation, condition 1 above, otherwise it would have to be vertical, and users would have to "tilt their heads" in order to see the graph properly. This not only would be very lazy, but it would be a terrible idea because:

  • There would be a huge, empty square between both graphs. The animation frame would be too large, and mostly empty space. This space would not be useful for anything else that wouldn't be conveyed better in the accompanying article or image description.
  • There would be no way to compare both graphs at once.

Therefore, his odd format is justified. Notice that this bend could be done either to the left or to the right. However, if to the right, the graphs would be drawn backwards in the animation, as they would be drawn from the left, and not to the right, as it is currently. This breaks condition 2, mentioned earlier.

I'm not sure if everyone would be OK with including this animation in the article. I couldn't figure where to place it anyway. So, for now, I'm just letting you guys know this animation exists. Cheers! — LucasVB | Talk 16:22, 16 March 2014 (UTC)

I've added this to the page—Love, Kelvinsong talk 16:05, 26 June 2014 (UTC)

"Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O"[edit]

I could not find this from the book, please provide proper referencing. — Preceding unsigned comment added by Bastasie (talkcontribs) 19:25, 23 April 2014 (UTC)

That part not changed, its without the reference for the figure below:

although its extremely familiar to me — Preceding unsigned comment added by Bastasie (talkcontribs) 16:19, 25 October 2014 (UTC)


I mean like, really, don't you feel this article could use some graphs, like of all the functions? Their absence is quite silly. I think there used to be some, what happened to them? Aoru (talk) 15:33, 26 June 2014 (UTC)

They are all condensed into one graph in the section Trigonometric functions#Unit-circle definitions. See File:Trigonometric_functions.svg. JRSpriggs (talk) 04:56, 27 June 2014 (UTC)

radians section is confused[edit]

The radians section is confused. There are real difficulties in the notion of "measure" of an angle, in the sense of associating a real number to an angle. Dieudonné has a careful discussion of this in his book LInear Algebra and Geometry, in which he concludes that it is impossible to measure an angle without making use of the complex exponential function. Another less explicit example of this is how most complex analysis textbooks use the power series definitions of cosine and sine to define the argument of a complex number, via the complex logarithm, and use that to measure angles.

A radian has to be dimensionless if you are going to plug it into a power series. Therefore this discussion contradicts the other sections of the article. The other sections are fairly careful to avoid saying that theta is a real number. If theta is an arclength, it has dimensions of length, but then theta squared has dimensions of length squared, so the power series expansion for sine makes no sense and certainly does not produce a ratio. In reality, angles are dimensionless, and the ratios of the sides of triangles are also dimensionless, so the other sections of this article succeed in avoiding the trap which this section falls into.

There is more than one way to fix this. Define radians as the dimensionless real number that makes sine satisfy the usual differential equation, or the usual Euler's formula, or the usual power series.

The arc length of the unit circle cannot be rigorously defined without using the integral calculus...actually, Jordan, in his Cours d'Analyse, is careful to define rectifiable curve and do enough integral calculus to define arclengths analytically right before deriving the derivatives of sine and cosine. Modern texts are usually not so careful and thus fall into a logical circle. Euclid was unable to study arclength as a real number, only areas, which is why it was left to Archimedes to introduce an extra axiom, about convexity, to study the arclength of the circle. (See Dieksterhuis on Archimedes, for example. Also Heath's commentaries.) (talk) 05:02, 1 September 2014 (UTC)

If Θ is an angle specified in any desired system (degrees, grad, fractions of a straight angle, or whatever), then
is that same angle in radians. So there is no problem defining it. JRSpriggs (talk) 10:26, 1 September 2014 (UTC)

(sin x)^2+(cos x)^2=1 =[edit]

when x is > or equal than 1 the following examples are true.I don't know if this is original research or not but it states that for all integers bigger than one, examples a)and b), the hypotenuse which faces the angle of right angle triangles is one, and 90 degrees for the angle and in radian 90 degrees is . I don't see these examples listed in article named trigonometric functions.

c) where is included in (talk) 21:42, 3 September 2014 (UTC)

Check my work[edit]

Although it borders on original research, I based this off the following work, which I believe to be rather straightforward (and the power series solution works because the tangent function is analytic around the origin):

The tangent function satisfies the differential equation with initial value , as mentioned in the article. Seek a power series solution in the form . Differentiate it once and substitute into the equation:

Apply the Cauchy product to the right hand side and subtract one from both sides:


Because (because of the initial condition):


Detach the first (k=1) term from the left side:


At x=0, the right hand side is zero and so are all the terms on the left side, which implies that (my justification of this is a bit shaky because this is only at one particular value of x; however is another way of justifying this value of that coefficient, albeit with what I see as an additional initial condition, which can of course be derived separately using the properties of sine and cosine). This just leaves

Equating terms of equal power gives


From this and the fact that we already found that , the terms I find match those given in the article:

To complete this assertion rigorously would require a proof by induction that the terms I find here match those given in the article, but it answers the question of how to algorithmically find the series directly as posed in an HTML comment in that section.--Jasper Deng (talk) 21:14, 13 January 2015 (UTC)

Circular reasoning in "significance of radians" subsection[edit]

In this section, an argument is presented which concludes that sine and cosine only obey the differential equations for sine and cosine when their parameter is measured in radians. The support for this claim, however, relies on the assumption that the parameter x in f(x) is in radians. If x is measured in degrees, and the derivative taken with respect to x, the differential equations for sine and cosine still hold. That is, there is nothing about the differential equation definition which fixes the dimensions of the parameter, as long as the parameter is a dimensionless quantity (radians or degrees both work perfectly well). I'll give this a week before I make any edits. Rangdor (talk) 00:50, 14 April 2015 (UTC)

The sine of an angle in radians and the sine of an angle in degrees are two different functions. To calculate the derivative of the sine, one uses the formula for the sine of the sum of two angles:
To get the desired result of cosine, we need:
The later result is only possible if h is given in radians, not degrees. JRSpriggs (talk) 10:51, 14 April 2015 (UTC)

Find a degree of a ratio without a calculator?[edit]

While not using a calculator, after learning how to find a non perfect square root (example: square root of 19 is in between 4 and 5, so as you keep using a number in between and squaring it until you are close enough to 19), I'm stumped on finding the degree value when you have opposite, adjacent, and/or hypotenuse. Here is an example:

sin(x) = 1/2

Solve for x without a calculator and/or prior knowledge (obvious, like avoid saying things like a square root of 100 is 10 without showing your work).

Joeleoj123 (talk) 12:35, 31 October 2016 (UTC)

In general, solving sin(x) = c to find x without using a calculator or trigonometric tables will be very difficult. But for the particular case sin(x) = 1/2 you can draw a right-angled triangle with hypotenuse length 2 and opposite side length 1, and then realize that this is one half of an equilateral triangle. Gandalf61 (talk) 12:46, 31 October 2016 (UTC)

Image placement[edit]

Several paragraphs of the section titled "Unit-circle definitions" talk about an "image to the right". However, at least on my computer monitor, the correspinding image is not placed to the right but actually on the next screen page (and even in the next section of the article). Can something be done about the placement of these images? – Tea2min (talk) 09:12, 17 February 2017 (UTC)

The article used the body of the article for providing information that belong to figure captions. I have solved the point by several edits for making the body of the section self-contained, moving the image that you mention to the top of the section, and using the same notation in the body and in the image (and other cleaning up that are not directly related to your complaint). Further work is still needed, in particular to create a section "Graphs", which is blatantly lacking. D.Lazard (talk) 12:00, 17 February 2017 (UTC)

External links modified[edit]

Hello fellow Wikipedians,

I have just modified one external link on Trigonometric functions. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

You may set the |checked=, on this template, to true or failed to let other editors know you reviewed the change. If you find any errors, please use the tools below to fix them or call an editor by setting |needhelp= to your help request.

  • If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
  • If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—InternetArchiveBot (Report bug) 21:38, 20 May 2017 (UTC)