Questions on trigonometric and hyperbolic functions, as well as their inverses, in Mathematica.

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26
votes
4answers
3k views

How to find the period of an arbitrary mathematical function?

Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
19
votes
2answers
845 views

ArcTan[0,0] numeric approximation

I've noticed that the numeric approximation of ArcTan is a bit different from the symbolic one. If I write ArcTan[0,0] I ...
17
votes
4answers
540 views

How do I introduce a new variable in a trigonometric equation?

I have the trigonometric equation \begin{equation*} \sin^8 x + 2\cos^8 x -\dfrac{1}{2}\cos^2 2x + 4\sin^2 x= 0. \end{equation*} By putting $t = \cos 2x$, I have \begin{equation*} \dfrac{3}{16} t^4+ ...
16
votes
1answer
438 views

Speeding up trigonometric integral

Context On a possible non trivial toric topology for the Universe (nothing less!). Problem I would like to carry out the following integral for $\ell=2,4\cdots 20$. $$\int _0^{\pi }\int _0^{2 \pi ...
15
votes
2answers
793 views

Why do certain values in ArcSin not fully evaluate?

This is a continuation of: Why do certain fractional values in TriangleWave not evaluate? The analysis by R.M and rasher revealed that problem to reduce to the behavior of: ...
14
votes
2answers
53k views

Generate a Unit Circle Trigonometry

Struggling often with Trigonometry I would like to have some code to generate this Unit Circle Trigonometry. Would be of great help when I need to transform some data :
13
votes
1answer
13k views

How to express trigonometric equation in terms of of given trigonometric function?

How can I express a trigonometric equation / identity in terms of a given trigonometric function? using following trigonometric identities ...
13
votes
2answers
778 views

Mathematica Sec and Csc

How can I prevent Mathematica from using the "old fashioned" functions "Sec" and "Csc"? In Germany these functions are "old fashioned" as they are not taught anymore at school.
12
votes
2answers
2k views

How do I work with Root objects?

I want to solve the trigonometric equation : $$(3-\cos 4x )\cdot (\sin x - \cos x ) = 2.$$ I tried Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x] It returns the ...
12
votes
4answers
495 views

Solve symbolically a transcendental trigonometric equation and plot its solutions

I currently have the equation below for a beam with a hinge at various locations, the variable a can vary between 0 and 0.5. $$ Tan\left( {a\sqrt \alpha } \right) + Tan\left\{ {\left( {1 - a} ...
12
votes
1answer
1k views

Persistence length of a polymer chain

I'm trying to calculate the persistence length (Lp) of a polymer chain using Mathematica. The first and simplest method is to compare the EuclideanDistance between ...
10
votes
2answers
2k views

Why doesn't Mathematica expand Cos[x]^3 Sin[x]^2?

I found some examples of Mathematica's commands usage in an old manual but the program gives me different result than expected ...
9
votes
2answers
3k views

How to expand tan(x+y) as normal form?

TrigExpand@Tan[x + y] gives $\frac{\sin (x) \cos (y)}{\cos (x) \cos (y)-\sin (x) \sin (y)}+\frac{\cos (x) \sin(y)}{\cos (x) \cos (y)-\sin (x) \sin (y)}$ but I ...
9
votes
1answer
269 views

Mathematica 10 unable to Simplify[Cos[2 ArcCos[A]]]

Mathematica 10 evaluates Simplify[Cos[2 ArcCos[A]]] as Cos[2 ArcCos[A]], instead of the desired ...
9
votes
3answers
571 views

Simplifying trig expressions containing Sqrt or Abs

The following Maple code can simplify abs(sin(x)) into -sin(x) given the condition ...
8
votes
4answers
304 views

How to make Mathematica simplify $\cos[(i + j + k)\pi]$ to $(-1)^{i+j+k}$?

We know that: $$ \cos\left[(i+j+k)\pi\right] = (-1)^{i+j+k} $$ for $i, j, k$ being positive integers. In Mathematica I've tried: ...
8
votes
1answer
6k views

Extract real part of a complex expression better than Re does

I have a complex expression with real positive variables only. Mathematica Input Style: ...
7
votes
4answers
2k views

Checking if two trigonometric expressions are equal

Say I have two trigonometric expressions which are a bit complicated. Is there a quick way to check if they reduce to the same thing (that they are equal) using Mathematica? I was solving this: $y'' ...
7
votes
6answers
864 views

Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$

Consider the function f[x_] := x Sin[Pi/x] I want to prove that this function is increasing for $x\ge 1$. This can be done with the first derivative. We have to ...
7
votes
2answers
381 views

How to automatically get rid of trig functions in an expression?

I have a large expression containing many terms like this: Sin[1/2 ArcTan[(2 Log[5])/(Log[5]^2 - 2)]] where a trig function is invoked on a rational multiple ...
6
votes
2answers
177 views

ArcTan and Compile

See following three snippet code: ...
6
votes
1answer
274 views

Simplifying a trigonometric sum

I want to caculate Cos[θ] + Cos[2 θ]+ ... + Cos[n θ] Sum[Cos[k θ], {k, 1, n}] ...
5
votes
4answers
561 views

How to transform the result of this integral into powers of $\sin\,x$?

I want to calculate the integral $I = \displaystyle\int{\cos ^5 x\, \mathrm dx}$. Using the variable $t = \sin x$, I get $$I = \sin x - \frac{2}{3}\sin^3 x + \frac{1}{5}\sin^5 x + C.$$ With ...
5
votes
4answers
722 views

Arcs of rotation for angles

How can I draw trigonometric angles of any radian measure associated with the unit circle, complete with the x-axis and y-axis? How can I draw arcs of rotation inside positive or negative angles? Is ...
5
votes
2answers
1k views

Solving a trig equation

I am trying to find an explicit solution to a trigonometric equation. It is running for a long time and I didn't get any result, but Maple solves it quickly. Why ? ...
5
votes
2answers
132 views

Apply Trig Identity to Transform Sum of Sinusoids into Phase-Shifted Cosine Function

I'm trying to solve for the output of a simple RC low-pass filter, with Cos[w t] input. The convolution of the input with the RC transfer function gives me the right output, but I'd like to be able to ...
4
votes
2answers
2k views

ArcTan expressed as a radian fraction?

This type of answer is what I'm looking for: In[58]:= ArcTan @ 1 Out[58]= π/4 This is what mathematica gives me: ...
4
votes
4answers
757 views

Simplifying a trigonometric expression

I'm sure this has been addressed before, but I can't find it! I want Mathematica to simplify Sqrt[1 - Sin[π/6 - θ]^2] to ...
4
votes
1answer
217 views

Eliminate 3 Variable from 4 Equations

I am trying to use Mathematica to Eliminate 3 following variable from 4 equations. Any clue or link to similar solved problem is highly appreciated: ...
4
votes
2answers
152 views

Why Refine does not simplify Cos[2*Pi*FractionalPart[1/2*(i + j + k)]] to $(-1)^{i+j+k}$?

I am obtaining in Mathematica: Cos[2*Pi*FractionalPart[1/2*(i + j + k)]] We know that for $i, j, k$ being positive integers this expression after simplification ...
4
votes
1answer
126 views

Infinite expression encountered when simplifying ArcTan sums although the result is finite

If I attempt to simplify the sum ArcTan[1/x] + ArcTan[x] with ArcTan[ Simplify[ TrigExpand[ Tan[ ArcTan[1/x] + ArcTan[x]]]]] ...
4
votes
2answers
262 views

Both Sin[x]==0 && Cos[x]==0 as a solution

If I do FullSimplify[Reduce[Sin[p1] == 0 && Cos[p1] == 0, Reals]] I get Cos[p1] == 0 && Sin[p1] == 0 ...
4
votes
1answer
415 views

Solving trig identities in specific terms

How would you solve a problem like "write csc(x) in terms of sec(x)" in Mathematica? The best I can get is "True." http://math.stackexchange.com/questions/167935/write-cscx-in-terms-of-secx I'm ...
4
votes
1answer
248 views

How to check if a given expression is an “explicit algebraic number”?

The documentation for PossibleZeroQ says: With the setting Method...
4
votes
0answers
81 views

Transform an expression to remove the singularity

I have the expression Sin[c x]/c. Clearly it is undefined at $c=0$, but that is merely a removable singularity, since $$\frac{\sin(cx)}c=x\frac{\sin(cx)}{cx}= ...
3
votes
3answers
203 views

Speed up Eliminate

I have a set of trig equations, c1,c2,c3 are constants, I excepted the result is c1 c2 c3 r^2 + (c1^2 + c2^2 + c3^2 - 1) r + c1 c2 c3 == 0 but following code ...
3
votes
4answers
222 views

Redefine ArcCot to be inverse of range [0, Pi]

Mathematica implements ArcCot function as an inverse to Cot on range [-Pi/2, Pi/2]. I ...
3
votes
3answers
302 views

How to extract phase angle from sinusoid

I'm doing some electric circuit calcualtions and I'm trying to get the phasor representation of some arbitrary function of Sin or Cos. Could be complex like: ...
3
votes
2answers
141 views

Mathematica not recognising closed forms as specific values

I have a Mathematica program that outputs a piecewise function, and some parts of the expression are not being recognised as specific numbers when using ...
3
votes
1answer
174 views

Mathematica takes too long to solve this

I have the following system of equations: ...
3
votes
2answers
520 views

Evaluating the integral of $\cos (\theta )\cos\left[\tan ^{-1}\left(\frac{a^2 \tan (\phi )}{b^2}\right)\right]$

I am trying to evalute the following integral: Integrate[Cos[θ] Cos[ArcTan[a^2/b^2 Tan[φ]]], {φ, 0, 2π}] I know that for $a=b$ I should get 0 out of the integral ...
2
votes
3answers
337 views

Problem with simplifying Sinc function

$Assumptions = x ∈ Reals && y ∈ Reals && a ∈ Reals && b ∈ Reals; These work: ...
2
votes
2answers
124 views

Is it possible to simplify a trigonometric expression only in terms of $\cos (u)$ OR $\sin (u)$?

I want to expand the following expression into powers of $\cos(u)$ only: a1 + a2 Cos[2 u] + a3 Cos[4 u] + a4 Cos[6 u] The answer that I want (and which I found ...
2
votes
2answers
152 views

Trigonometry expansion of a product

The following command does what I want: ...
2
votes
3answers
385 views

DSolve will not apply assumption m ∈ Integers

I am trying to solve a linear second order ODE using DSolve which involves an arbitrary integer m. ...
2
votes
1answer
439 views

Solve[Tan[theta] == (b*Sin[t])/(a*Cos[t]), theta] breaks when “Reals” is added?

Mathematica solves this equation fine: Solve[ Tan[theta] == (b*Sin[t])/(a*Cos[t]), theta] // InputForm ...
2
votes
1answer
284 views

Problem with Sinc[x] function and rounding

I seem to be having some trouble with the Sinc[x] function, and I think it's due to rounding, but I'm not sure. The code is below, with relevant constants: ...
2
votes
2answers
320 views

How do I get the solutions of a cubic equation in trigonometric form?

I know $x^3-3 x+1=0$ has three roots that can be expressed in trigonometric form: $\{2\sin(10^\circ),\,-2\cos(20^\circ),\,2\cos(40^\circ)\}$. How can I get this result with Mathematica?
2
votes
1answer
153 views

Discard functions with wrong symmetry?

I have functions of x, where x is the polar angle of spherical coordinates, i.e. ...
1
vote
4answers
1k views

Solve trigonometric equation

I'm looking for a solution (graphic, numeric or analytic) for a function of the form: $$a^2\tanh(x)=\tan(ax),\quad a ≥ 1,\ x > 0$$ preferably using Mathematica. Any advice?