# Talk:List of trigonometric identities

WikiProject Lists (Rated List-class)
This article is within the scope of WikiProject Lists, an attempt to structure and organize all list pages on Wikipedia. If you wish to help, please visit the project page, where you can join the project and/or contribute to the discussion.
List  This article has been rated as List-Class on the project's quality scale.
WikiProject Mathematics (Rated List-class, Mid-priority)
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:
 List Class
 Mid Priority
Field:  Geometry
One of the 500 most frequently viewed mathematics articles.

## More than two sinusoids

As in the "Arbitrary phase shift" and "Sine and cosine" cases the expressions for a and θ should be unambiguous. To the best of my knowledge that is

${\displaystyle a={\sqrt {\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}}}$

and

${\displaystyle \theta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \theta _{i},\sum _{i}a_{i}\cos \theta _{i}\right).}$

## Product-to-sum and sum-to-product identities

These formulas have a name (a classical name). Unfortunately I don't remember exactly the spelling. The name is (or approximately is) prostospheratic formulas or prostoferatic formulas. I know it is a not commonly used name these days but it would be nice if someone remembers it/ finds it and finds even why they were called that way. I am asking this not only for making an addition to thee article but also for myself. I want to remember the name and learn the neaning/ origin of it. Thanks.  franklin  03:53, 26 January 2010 (UTC)

I suppose that might be mildly interesting, but it seems to me that any such names would be too obscure to be of much encyclopedic interest. JamesBWatson (talk) 08:24, 28 January 2010 (UTC)
• Not what I am asking. I am just asking for the name not inviting to any analysis of its encyclopedic value. By the way, the name (the meaning of the Greek origin) gives some geometric interpretation or some application of the formula. But for the moment I just want to remember the name the rest depends on what exactly the name was.  franklin  11:41, 28 January 2010 (UTC)
I believe Prosthaphaeresis is what you're looking for. Dmcq (talk) 11:46, 28 January 2010 (UTC)
• Thanks so much. I knew the name in Spanish but no way I could have guessed all those h's in the name in English. And it seems to be that the name had EV after all since it already have an article. So, a link from here to that article would be useful.  franklin  11:52, 28 January 2010 (UTC)
Hope someone reads this, I do not really post on wiki often, but wanted to get this info out there (I'm sure I'm breaking some rule somewhere but I thought this was important to point out); for the SINX±SINY sum-to-product rule I have that the SIN and COS are switched in the result, not the ± symbols. The way I read it here suggests that if you have sinx+siny then the 2sin(1/2*x+y)cos(1/2*x-y) and if it is sinx-siny then you get 2sin(1/2*x+y)cos(1/2*x+y), thinking only that the symbol (+ or -) relating the two angles of COS are changed in the result. I'm posing that if sinx+siny then 2sin(1/2*x+y)cos(1/2*x-y) or if SINx-SINy then 2COS(1/2*x+y)SIN(1/2*x-y), only the sin and cos are switched, everything else stays the same. Am I totally off? Info from a calculus MathXL online course (precalc review chapter) problem GR.4.111. I don't know the html tags to make the time i worte this and all that stuff official :( sorry. you can delete this or revise it to fit in better, assuming my comment made any sense (or is even valid). nagromltNagromlt (talk) 21:40, 16 March 2011 (UTC)

I believe that there is an error in these identities, at least the first sum-to-product identity; they should not have both (+) and (-) operators in each function. The cosine function should only have the (-) sign while the sine function should only have the (+) sign. Without demonstrating a proof here is a quicker way using listing of a simple GNU Octave (Matlab) script that clearly shows it. Note that when the operators are reversed it results in a phase shift of the resulting waveform. CODE LISTING :

%GNU Octave script for demonstrating sum to product Identity
clear all; close all

%.............. Create independent variables
t = 0:0.01:100; %first variable
t1 = t.*0.9; %second variable

%.............. Left side of identity .....................
A = sin(t); %first sinusoid
B = sin(t1); %second sinusoid
C = A+B; %superposition

%.............. Right side of identity ....................
E = cos((t-t1)./2); %cosine subtract
F = 2.*(D.*E);

G = sin((t-t1)./2); %sine subtract
J = 2.*(G.*H);

%.............. Plot waveforms .............................
figure %figure 1 sine add, cosine subtract
plot(t,C,t,F) %note that these waveforms are indeed identical

figure %figure 2 sine subtract, cosine add
plot(t,C,t,J) %note that these waveforms are NOT the same

63.231.234.97 (talk) 17:11, 1 February 2012 (UTC)

Oops, above entry entered in error please disregard since I failed to simultaneously change the operator in the left side of the equation.63.231.234.97 (talk) 17:36, 1 February 2012 (UTC)

## Triple tangent/cotangent identities

Perhaps my edit of 04:43, 23 November wasn't clear. I believe that, if it is acceptable to give these things names, the way I have done it is correct. I don't understand the objection that "changing the cotangent identity to the sine-double identity (even if accurate) is not [acceptable]". It wouldn't be accurate, and I never meant to imply such a thing. It is not my intention to give the equation about tangents any name other than the tangent identity, the one about cotangents any name other than the cotangent identity, or the one about sines any name at all. I'm not aware that the one about double sines has any name, though if someone comes up with one, that's fine. SamHB (talk) 16:58, 23 November 2013 (UTC)

## Proof of Chebyshev formulae

Just in case it's useful to anyone, the Chebyshev formulae for ${\displaystyle \sin nx}$ and ${\displaystyle \cos nx}$ are most easily proved as the imaginary and real parts (respectively) of the formula:

${\displaystyle \operatorname {cis} nx=2\cdot \cos x\cdot \operatorname {cis} ((n-1)x)-\operatorname {cis} ((n-2)x)}$

where ${\displaystyle \operatorname {cis} x=\cos x+i\sin x=e^{ix},}$ as in Euler's formula. Because this is an exponential function, it has the particularly simple summation function ${\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} x\operatorname {cis} y}$, and ${\displaystyle \operatorname {cis} nx=\operatorname {cis} ^{n}x.}$

Given this, the formula can be simplified by dividing both sides by ${\displaystyle \operatorname {cis} ((n-2)x)}$ to get:

${\displaystyle \operatorname {cis} 2x=2\cdot \cos x\cdot \operatorname {cis} x-1.}$

Which can be proved by expanding and then simplifying the right-hand side:

{\displaystyle {\begin{aligned}2\cdot \cos x\cdot \operatorname {cis} x-1&=2\cdot \cos ^{2}x+2i\cdot \cos x\cdot \sin x-\cos ^{2}x-\sin ^{2}x\\&=\cos ^{2}x+2i\cdot \cos x\cdot \sin x-\sin ^{2}x\\&=(\cos x+i\cdot \sin x)^{2}\\&=\operatorname {cis} ^{2}x\\&=\operatorname {cis} 2x\end{aligned}}}

71.41.210.146 (talk) 17:06, 28 December 2013 (UTC)

## Symmetry

Forgive me if I'm being an idiot, but in the table that includes the cofunction identities, should the transformations not be described as reflections in ${\displaystyle \theta =0,\theta ={\pi \over 4}}$ and ${\displaystyle \theta ={\pi \over 2}}$ respectively? M.A.Redman (talk) 19:14, 10 April 2014 (UTC)

## useful identity

It would be handy to give a solution to the geodesic eqn ${\displaystyle a\sin x+b\cos x=c}$ 67.198.37.16 (talk) 20:01, 30 September 2015 (UTC)

## If the article gets too long, how shall we reorganize or split it?

By WP:SIZESPLIT and Wikipedia:Article size pages above 60 kb size should be split List of trigonometric identities is about 71 kb so is in need of a split.

Also the list is ab bit hard to follow so lets have a discussion about how we reorganize this page and maybe if and how how we should split it.

My idea is to split it in two, one with let us call it the high school trigonometry and a second one with more advanced subjects.

Advanced subjects would for me include:

and many more lets discuss before we edit that makes nobody happy. WillemienH (talk) 12:44, 25 February 2016 (UTC)

I think the article is on a coherent single topic and so should not be split up. I don't think it's hard to navigate, since there's a table of contents. With two separate articles, someone might have a hard time finding what they're looking for, if they go to the wrong article. Loraof (talk) 20:31, 5 June 2016 (UTC)
I agree. I don't think splitting up is a good idea. The cited guidelines with the 60kb are more than a crude rule of thumb than an absolute figure. We have plenty of compilation and overview articles that are larger than 60kb and where splitting them up is simply not a good option. In fact many of our excellent articles are > 70kb.--Kmhkmh (talk) 20:53, 5 June 2016 (UTC)
I don't know if splitting is needed but holy crap is this page difficult to read with the current format and organization. — Preceding unsigned comment added by Bofum (talkcontribs) 04:44, 1 October 2018 (UTC)

## Pythagorean Identity

When I learned the following identities, they were all called "Pythagorean identities" with consistency from teacher to teacher:

${\displaystyle \sin ^{2}x+\cos ^{2}x=1}$
${\displaystyle \tan ^{2}x+1=sec^{2}x}$
${\displaystyle 1+\cot ^{2}x=\csc ^{2}x}$

However, online it seems that some people only use the term "Pythagorean identity" to refer to the former. Therefore, would anybody object to me changing "identity" to "identities"? Thank you.LakeKayak (talk) 19:18, 18 February 2017 (UTC)

As it seems that nobody has an opinion, I am going to make the change and wait for people's responses from there.LakeKayak (talk) 01:28, 21 February 2017 (UTC)