# Questions tagged [trigonometry]

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

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### How to force Mathematica to simplify in terms of half angles?

By basic trigonometric identities, the expression $\frac{\sinh(x)}{\sqrt{2}\sqrt{1+\cosh(x)}}$ simplifies to $\sinh(\frac{x}{2})$. But Mathematica seemingly don't perform this simplification. I tried ...
75 views

### Mathematica's definition of cos(kA) where A is the 1st Pauli spin matrix [closed]

For a homework assignment I was asked to derive the formula of $e^{ik\sigma_{1}}$ - where $\sigma_{1}$ is the Pauli spin matrix {{0,1},{1,0}} - using the Taylor ...
46 views

### how to get the set of coefficient of a trig function in terms of the sines and cosines of a given angle

Say I have an expression like this a Sin[\[Theta]]^4 + b Cos[\[Theta]]^4 + c Sin[\[Theta]]^3 Cos[\[Theta]] + d Sin[\[Theta]]^2 + e=0 My ultimate goal is to solve ...
249 views

### Drawing angle bisectors in a triangle

In triangle $\triangle ABC$, the angle $\angle BAC$ is equal to $60°$, $|AB|=2$, $|BC|=\sqrt 6$, and $AD$ bisects the angle $\angle BAC$ and $BC$ at point $D$. How can I find the length of the angle ...
96 views

### Reduce and Solve give incorrect answers for trigonometric equation [closed]

Why use Reduce and Solve to solve the Trigonometric functions equation with incorrect results? ...
73 views

### Find the interval that satisfy the equations

I am having issues solving rather simple equations with these functions. The equations are I have plotted each equation and its obvious that there are multiple solutions to these equations (...
152 views

### Simplify solutions of trigonometric equation has the form Sin[a]==Sin[b]

I am trying to solve the equation $$\sin\left(2 x + \dfrac{\pi}{5} \right) = \sin \left(3 x-\dfrac{2 \pi }{7} \right).$$ My code ...
111 views

### Simplify solution of trigonometric equation

I use this code to solve equation Sin[2x]==0] Reduce[Sin[2 x] == 0, x] C ∈ Integers && (x == π C || x == 1/2 (π +...
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### How can I draw a triangle and find the length of its height?

Given that the sum of three inner angles $a,b$ and $c$ in a triangle is $\pi$, and knowing that： $$2 \sin(a−c) = \sin b,$$ $$a+b=3c,$$ how can I draw this triangle and find the length of the height on ...
38 views

### Invalid Variable Error for Solve When Using Law of Cosines

I'm making a shadow simulator, and am using the Law of Cosines to calculate a certain angle in a triangle with two known widths and one known angle. I'm using Solve ...
68 views

### 3D plot not working, find max/min points [closed]

Im trying to plot this graph in mathematica: (x+y)sin(x-y) And this is my in- and output: Plot3D[(x + y) (Sin[x - y]), {x, -50, 50}, {y, -50, 50}] The output is ...
1 vote
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### How to find the maximum value of a parameter [Omega] in a trigonometric function?

The form of the known trigonometric function is: f[x_] := Sin[ω x + ϕ] The known conditions are: ω > 0, ϕ ∈ {0, 2 π} One of ...
460 views

### How to find the maximum value of this trigonometric function?

The known conditions are: Sin[2 α + β] == 2 Sin[β], 0 < α < π/2, 0 < β < π/2 Find the maximum value of Tan [β] <...
1 vote
173 views

### How to solve this trigonometric function equation?

Solve[{TrigExpand[Tan[2 α]] == Cos[α]/(2 - Sin[α]), tanα == Sin[α]/Cos[α], 0 < α < π/2}, tanα, {Sin[\[Alpha]], Cos[\[Alpha]]}] It returns <...
176 views

### How can expanding trigonometric function retain an expression?

I want to expand this trigonometric function Sin[π/4 + α - β] such that $\alpha - \beta$ is treated as "one" expression, resulting in ...
1 vote
78 views

### Solving a trigonometric equation for the angle in radian

I have been trying to solve the following equation in radians: NSolve[0.6 Cos[π/6 + 2. x ] + Sin[x] == 0, x] Following this, I can modify the code as ...
1 vote
79 views

### Solve equation - Error - Help

I need to find the value of n but I get an error. I used this: ...
70 views

### How to switch Tan[x] for Sin[x]/Cos[x] without Mathematica changing the latter for the former? [duplicate]

When I ask Mathematica to do this: (Sin[2 x] - Tan[2 x])/x^3 /. Tan[x_] -> Sin[x]/Cos[x] It makes no change because Mathematica interprets ...
145 views

### Evaluate using Mathematica or otherwise $\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx$

Denote $T_n(x)$ as Chebyshev polynomial of the first kind (see here). Then I need to evaluate for $n$ a odd natural number $$\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx$$ I am requesting a code with ...
1 vote
45 views

### Expression of trigonometric function

I have a complicate expression with a lot of trigonometric functions. And I want to express the function just in the form of $\cos(\theta)$ and $\sin(\theta)$ instead of something like $\cos(2\theta)$ ...
89 views

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### Given the trigonometric value of the double angle, how to find the trigonometric value taught by double？

Known this Tan[2 A] == 2 Sqrt How do you evaluate its value? Tan[A] ==? Sin[A]==? Cos[A]==?
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### Is there an option to automatically make solve return the principal value of inverse trig functions?

I am porting some of my Maple packages to Mathematica. In Maple, solve(sin(y)=x,y) returns arcsin(x) and I'd like to get same ...
629 views

### How to calculate the integral only in real domain?

I have the following integral: Integrate[1/(Sqrt + Cos[4 X] + Sin[4 X]), X, Assumptions -> Element[X, Reals]] // Simplify ...
103 views

### How can I make sure that the two given numbers are exactly the same?

I have the given numbers $n1$ and $n2$. How can I make sure that these two numbers are exactly the same? Numerically, using {N[n1,40], N[n2,40]}, by increasing the ...
158 views

### Solving a polynomial in trigonometric functions

Given all variables are in $\mathbb{R}$, and $\{ a, d\} \neq 0$, solve for $\theta$: $$a \sin^2 \theta + b \sin \theta + c + d \cos^2 \theta + e \cos \theta + f = 0$$ as posed in this question. The ...
1 vote
83 views

I have a trigonometric function $Exp$ where $\{f,g,h,k\}$ are some parametric functions, and $a\in\mathbb{N}$ and $x>0$. $$Exp=f \cdot \sin \left(\frac{58 \pi a}{11}+x\right)+g\cdot \sin \left(\... 0 votes 0 answers 66 views ### How does this trigonometric simplifier work? I recently watched this video of Wolfram channel about modelling black holes with Mathematica. https://www.youtube.com/watch?v=bOtvvw1dyN0 At 13:45 In it, they have created a trigonometric simplifier, ... 1 vote 0 answers 62 views ### Simplification problem using trigs given f1 = Sin[x]/Sqrt[2 + Sin[2 x]] f2 = (Cos[x]-Cos[3 x]+4 Sin[x])/(2 (2+Sin[2 x])^(3/2)) These are same for all x. ... 5 votes 3 answers 220 views ### Collect the coefficients through the Cos argument I have for example the following expression: ... 2 votes 1 answer 94 views ### Slow integration of trigonometric expression speeds up after TrigReduce I'm trying to integrate a trigonometric expression as Integrate[Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t], {t, 0, 2 Pi}] The final results is correct (... 1 vote 0 answers 46 views ### Power of trigonometric function matrix [closed] Calculate the m-power of the following trigonometric function matrix (m>0): \boldsymbol{A}=\left(\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{array}\... 4 votes 1 answer 86 views ### How can I have Mathematica solve the curve equation when nested functions are involved? In the context of Quantum Computing, I am investigating curves that illustrate an error propagation during qubit rotations on a Blochsphere (I asked a related question in the same context here and ... 3 votes 0 answers 129 views ### Cannot Understand nth Derivative of x/ArcTan[x] The nth derivative of x/ArcTan[x]: f[x_, n_] = D[x/ArcTan[x], {x, n}] Evaluates to: I cannot get this general from to return ... 3 votes 2 answers 131 views ### Replacing a trigonometric value It looks like MMA cannot evaluate Tan[42°] but it can be written as (-Sqrt[6 - 2 Sqrt] + Sqrt[6 (5 + Sqrt)])/(2 Sqrt[7 - ... 0 votes 1 answer 85 views ### Unable to simplify exponential expression to its simplest form of trigonometric expression by Euler's formula Applying "FullSimplify" fails to simplify the following expressions: Input: ... 1 vote 1 answer 156 views ### convert n*Arctan[x] to ArcTan[x] In the result of solving the equation or the integral often get something like k*ArcTan[x], I prefer the ArcTan[x] form. For ... 4 votes 3 answers 292 views ### How to build an inscribed circle in a triangle on a sphere? I want to build an inscribed circle into a triangle on a sphere. To do this, I first build its center - the intersection point of the bisectors. But there were problems with finding it. I build the ... 1 vote 1 answer 171 views ### Solving multivariable trig Identities I'm trying to solve for a certain set of trig identities in terms of other trig identities. I saw this and this, but in my example, I have two tries: ... 2 votes 1 answer 81 views ### Reduce yields ArcTan instead of Pi/4 I expect this call: Reduce[Sin[x] - Cos[x] == 0, x, Reals] to return x = \pi/4 + 2k\pi \ \vee \ x = -3/4 \ \pi + 2k\pi. However, \pi/4 and 3/4\ \pi get ... 7 votes 2 answers 279 views ### How to pose simultaneous trigonometric equations problem in a way solvable by mathematica Context: A camera at O has an image of a rectangular object ABCD. From the 2D coordinates of the image + the projection matrix of the camera I can calculate the direction vectors \hat {A}, \hat{B}, \... 0 votes 0 answers 107 views ### Simplification of long trigonometric expression taking a long time I have tried to Simplify as well as FullSimplify a long trigonometric expression to get a simplified form of the expression but it takes a very long time almost hours and sometimes it also shows error ... 2 votes 1 answer 84 views ### How to do some trigonometric derivations Two substitutions fail, so do I forget some earlier steps? For Cos[x] == Sin[(1/2)*Pi + x] /. x -> (1/2)*x = true I don't get the final answer, but only true. <... 9 votes 6 answers 608 views ### Finding which given functions are equal to each others I have about 50 piece functions (generally trigonometric functions). I want to find out which ones are equal to each other after some simplifications such as ... 4 votes 2 answers 178 views ### Solving a set of trig inequalities I have a set of inequalities Cos[a]Cos[b]>=Cos[t-a]Cos[b]&&Cos[a]Cos[b]>=Cos[t/2]&&Cos[a]Cos[b]>=Sin[t/2]&&a<=t<=Pi How to ... 4 votes 1 answer 247 views ### Mathematica vs Rubi --- integrate f(x,y) = x \sin^2 x + a x y --- Mathematica got this round? [closed] Consider the function:$$ \begin{equation} f(x,y) = x \sin^2 x + a x y \end{equation}  with $a$ being a constant. We want to do the indefinite integral over $x$ and then over $y$. We do that using <... 1 vote
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### How to sort out only terms that relates to certain variable in the results of TrigReduce? [closed]

I have a large function $F(\theta,\alpha_1,\alpha_2,...)$ which involves only trigonometric polynomial of these angles. I want to calculate $\frac{1}{2\pi}\int_0^{2\pi} Fd\theta$, but the function is ...
546 views

### How do you make Tan[A+B] expansion give the well known high school answer? [duplicate]

What command makes Tan[A+B] expand to (Tan[A]+Tan[B])/(1-Tan[A]Tan[B])? TrigExpand does not ...
I found something odd in Mathematica when using the SphericalPlot3D command. Ranges for $x,y∈[0,2 \pi]$ are the same. All I do is swap sine and cosine. But when I ...
### How can I get mathematica to prove $\sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right)$?
How can I get Mathematica to prove $\sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right)$ (1)? ...