Questions tagged [trigonometry]

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

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41
votes
3answers
12k 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 ...
31
votes
1answer
29k 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: ...
30
votes
2answers
61k 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 ...
29
votes
4answers
6k 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?
25
votes
1answer
828 views

Transform Root objects into Trigonometric expressions

Consider the Root objects roots = Table[Root[-1 + 27 #1^2 - 162 #1^4 + 243 #1^6 &, i],{i,1,6}] These can be expressed in terms trigonometric functions as ...
23
votes
3answers
4k views

Behavior of ArcTan when two zeros are given as arguments

I've noticed that the numeric approximation of ArcTan is a bit different from the symbolic one. If I write ArcTan[0,0] I ...
21
votes
3answers
3k 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.
18
votes
2answers
87k 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 :
18
votes
4answers
710 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+ \...
18
votes
1answer
677 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 }...
17
votes
5answers
1k views

Laplace transform of $\frac{1-\cos (t)}{t}$

In the documentation, it states that The Laplace transform of a function $f(t)$ is defined to be $\int_0^{\infty } f(t) e^{-s t} \, \mathrm{d}t$. But why can Mathematica not get the Laplace ...
17
votes
5answers
3k 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} \right)\...
17
votes
2answers
2k 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
1answer
3k views

Simplify Sin[x]/x to Sinc[x]

I have an expression in the form of $\tt \frac{Sin[x]}{x}$ that I would like to simplify to the form of Sinc[x]. I've tried the ...
14
votes
3answers
4k views

Combining cosine or sine terms into a single cosine or sine

A function that is the sum of two cosine terms of the same frequency x[t_] := a1 Cos[w0 t + b1] + a2 Cos[w0 t + b2]; can be simplified to a single ...
14
votes
2answers
379 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}= x\...
14
votes
1answer
370 views

Factorize trigonometric matrices

Consider two square matrices $A_1$ and $A_2$. Consider the following matrix involving matrix trigonometric functions: \begin{equation} M_1(t)=\begin{bmatrix} \cos(tA_1) & t\mathrm{sinc}(t A_1) \\ ...
13
votes
1answer
2k 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 ...
13
votes
0answers
107 views

Reduce[Sin[x]==0,x] splitting odd and even multiples of Pi

as from Title, I'm wondering why Mathematica does not automatically give a single, simple solution for the problem Reduce[Sin[x] == 0, x] for which I obtain the ...
12
votes
1answer
600 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 ...
11
votes
4answers
6k 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'' ...
11
votes
7answers
3k views

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

As an intermediate step, 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 ...
11
votes
2answers
6k 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 ...
11
votes
3answers
317 views

Simplifying Cosh[a] + Sin[a] + Sinh[a]

Why does FullSimplify[Cosh[a] + Sinh[a] + Sin[a]] not return Exp[a]+Sin[a] ? Is this some bug in Mathematica or am I being ...
11
votes
1answer
332 views

How does Mathematica calculate $\sin(\pi/5)$?

Consider: Sin[Pi/5] which returns: $$\sqrt{\frac58-\frac{\sqrt5}{8}}$$ Does anyone know how to figure out the trigonometric identities used by Mathematica to ...
10
votes
2answers
591 views

Evaluate trigonometric function

Can anyone help me to evaluate the trigonometric function below? Please look that the picture for more details. Here is the code to evaluate: ...
10
votes
3answers
2k views

Simplifying trig expressions containing Sqrt or Abs

The following Maple code can simplify abs(sin(x)) into -sin(x) given the condition ...
9
votes
4answers
595 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: ...
9
votes
2answers
729 views

How to solve this system of trigonometric trancendental equations over the reals?

Namely, $$\left\{x^2+2 x \sin (y)+3 \cos (y)=0,\sin ^{-1}\left(\frac{x}{2}+\sin (y)\right)=y-\frac{\pi }{3}\right\}$$ My simple-minded trial ...
9
votes
1answer
754 views

Need help on the proper use of a formula to produce a cambered airfoil

Following the book on "The Theory of Thin Wing Sections" page #112 describes the method to combine a camber line (i.e Mean Line) and a thickness distribution to form a cambered wing section. Data ...
9
votes
2answers
7k 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
302 views

Why ArcTan[1, 0. I ] yields -1.5708+0. I?

Bug introduced in 5.2 or earlier and persisting through 12.0.0 Why ArcTan[1, 0. I ] yields -1.5708+0. I ? The result should ...
8
votes
2answers
135 views

Computing the seven roots of a polynomial

This question was originally asked by @fsrong70 six months ago. The OP deleted it shortly after posting and has not returned to this site since. I had just figured it out when it was deleted. I ...
8
votes
2answers
156 views

Numerical solve question

Bug introduced in 9.0 or earlier and fixed in 10.4.0 Why does this work? ...
7
votes
5answers
1k views

Fitting a Cos^2 function

I have to fit a Cos^2 function to data I measured. The function is $a \cos^2(\frac{bx\pi}{180}+c)+d $ and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the ...
7
votes
2answers
268 views

Can I get Mathematica to recognize common series expansions of trigonometric functions?

I have some complicated functions from numerical procedures that go to simple trig functions in the limit of small time-steps. For example, I can calculate that one of these functions goes to: ...
7
votes
2answers
1k 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 of ...
7
votes
1answer
389 views

Integral that yields ArcTan function rather than ArcSin

When I enter the following Integral problem to Mathematica, $$ \int \frac{1}{\sqrt{a^2-x^2}}\textrm{d}x $$ The answer yielded by Mathematica is $$ \arctan\left({\frac{x}{\sqrt{a^2-x^2}}}\right) $$ ...
7
votes
3answers
172 views

Finding points in a plane

The end of the Star Wars saga is near, so I feel compelled to design a Mouse Droid. Based upon the Paul Murphy's schematics I've reconstructed the outline of the top shell: Since my goal is to ...
7
votes
1answer
298 views

Quantity and Trig functions

I have a bunch of data to which I've assigned units using Quantity, including some angular measurements in degrees. I'd like to be able to use these measurements ...
6
votes
3answers
2k views

Replacing Tan[x] with Sin[x]/Cos[x]

Why doesn't this work? (1 - Tan[x])/(Sin[x] - Cos[x]) /. Tan[x] -> Sin[x]/Cos[x] I get back: ...
6
votes
5answers
788 views

How can I make Mathematica do, ArcTanh[x] + ArcTanh[y] = ArcTanh[x+y/1+xy]?

I run this code: Simplify[ArcTanh[x] + ArcTanh[y]] and it keeps returning: ArcTanh[x] + ArcTanh[y] How do I direct it to ...
6
votes
4answers
918 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 ...
6
votes
2answers
267 views

ArcTan and Compile

See following three snippet code: ...
6
votes
2answers
1k views

Mathematica in degrees mode

How can I tell Mathematica to evaluate all expressions, plot all functions, solve all equations in degrees, instead of radians?
6
votes
4answers
196 views

Applying `TrigToExp` only for terms containing certain variables

Consider a Minimum Working Example, expr = Sin[a] (-I Cos[b] + Sin[b]), is there some way to only apply TrigToExp[] for ...
6
votes
2answers
419 views

Evaluating complicated trig without resorting to numerical methods

I have an angle theta given by Kx = (2/Sqrt[3]) ArcSin[Sqrt[3/10]]; Ky = (2/3) (ArcCos[(1/5) Sqrt[7/10]] - Pi); theta = Cos[ArcTan[Kx, Ky]]; This evaluates, by ...
6
votes
1answer
167 views

Solve can't solve my trigonometric equation—why?

I have the following equation for epsilon $2 |\alpha_{n,ch}|=\epsilon+\cos^{-1}(\tan(\frac{\pi}{4}-\frac{\epsilon}{2}))$, where $\alpha_{n,ch}$ is a negative constant (e.g. -0.769) I tried to solve ...
6
votes
2answers
551 views

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

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 ...
6
votes
2answers
117 views

Volume integration with a condition

I have a set of parameters $x_1, \dots x_4 \in (0,1)$. I want to find the volume of the subregion of this 4d cube satisfying $$\sin^{-1}(x_1)+ \dots +\sin^{-1}(x_4) > \frac{3\pi}{2}$$ I've tried ...

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