Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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2answers
21 views

What is the period of these functions?

I have two functions as follows: $x = (a-b) \cdot \cos(t) + b \cdot \cos(t\cdot(k-1))$ $y = (a-b) \cdot \sin(t) - b \cdot \sin(t\cdot(k-1))$ What are the periods of functions $x$ and $y$? I found ...
2
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2answers
52 views

$|\sin(\sin( \cdots \sin(x)\cdots))|$ ($N$ times) is always $\leq|\sin( \cdots \sin(1)\cdots)|$ ($N-1$ times)

Is this inequality always true? $$ \bigl\lvert\,\underbrace{\sin(\sin(\cdots \sin}_{N\text{ times}}(x)\cdots))\bigr\rvert\le\bigl\lvert\,\underbrace{\sin(\sin( \cdots \sin}_{N-1\text{ ...
1
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1answer
21 views

Prove $ \frac{4cos^{2}(2x)-4cos^2(x)+3sin^2(x)}{4cos^2(\frac{5\pi}{2} - x) -sin^22(x-\pi)} = \frac{8cos(2x)+1}{2(cos(2x)-1)}$

Question: Prove $$ \frac{4cos^{2}(2x)-4cos^2(x)+3sin^2(x)}{4cos^2(\frac{5\pi}{2} - x) -sin^22(x-\pi)} = \frac{8cos(2x)+1}{2(cos(2x)-1)}$$ My attempt starting with the bottom ...
0
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2answers
39 views

History and origin of sine function

I'm doing some research about the beginning of trigonometry. I want to know why and who draw the first time the sine function. Do you have one site or something that can help me ?
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2answers
35 views

Prove the inequality regarding complex numbers

If $\theta_i\in [0,\pi/6],i=1,2,3,4,5$.And $$\sin \theta_1\ z^4 + \sin\theta_2 \ z^3 + \sin\theta_3 \ z^2 + \sin\theta_4 \ z + \sin\theta_5=2$$ Prove that $|z|\gt \frac{3}{4}$.
0
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0answers
13 views

offsetting 90 degree of a line equation

i'm writing an application and i need to duplicate a line and offset it 10 pixel in 90 degree, how can i do that? detail: okay, let say i have: $P_1(x_1=0,y_1= 4)$ $P_2(x_2=5,y_2= 4)$ and equation ...
1
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4answers
63 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
0
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2answers
19 views

Finding $2$ points from angle.

I have $2$ points, $p_1(x_1=0,y_1=0)$ $p_2(x_2=5,y_2=5)$ And if i want to know what angle these $2$ points make. I can say, since $\sin \theta$ is $y$ axis and $\cos \theta$ is $x$ axis, so i ...
1
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2answers
37 views

Third Ailles Rectangle

The Ailles rectangle (named after an Ontario high school teacher, D. S. Ailles) is a rectangle of size $\sqrt{3}\times\sqrt{3}+1$ with three kind of triangles, like below. We have triangle ...
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0answers
24 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
0
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1answer
30 views

Water main construction. Find the angle using vectors.

A water main is to be constructed with a $12.5$​% grade in the north direction and a $25$​% grade in the east direction. Determine the angle $\theta$ required in the water main for the turn from north ...
1
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1answer
19 views

Proof of Trigonometric Equation with using Complex Numbers

Prove this identity without using complex numbers: $$P(z, t) = A \cos(ωt -Bz + θ_1) + D \cos(ωt -Bz + θ_2) = C \cos(ωt -Bz + θ_{total})$$ Where $C = \sqrt{(A)^2 + (D)^2 +2AD\cos(θ_1 - θ_2)}$ and ...
6
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4answers
62 views

Showing Trigonometric Identity

Prove that: $$\cos^2\theta\sin^4\theta=\frac{1}{32}(\cos6\theta-\cos2\theta+2-2\cos4\theta)$$ Attempt: \begin{align*} L.H.S & = \cos^2\theta\sin^4\theta\\ & = ...
2
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1answer
54 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
0
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0answers
15 views

Prove that the area of a triangle DEF is correct.

There's any triangle ABC. First player 1 has to set D on AB so that in the end the triangle DEF has the highest possible area. Second player 2 has to set E on BC so that in the end the triangle has ...
1
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2answers
35 views

Triangles - sin, cos etc. [on hold]

I know this is a quite simple question for most of you out there. However it has been a little troubling for me, and would like to get a little help if possible. I have a triangle $ABC$ where I know ...
1
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1answer
47 views

How do I find the difference between the gradients of two lines represented by an equation

I want to find the difference between the gradients (or slopes?) of two lines. The equation of the lines is $$x^2(\tan^2 \theta+\cos^2 \theta)-2xy\tan\theta+y^2 \sin^2 \theta=0$$ I have assumed the ...
2
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2answers
45 views

Roots of trigonometric function.

I have a function: $ f(x) = (ax+b) \cdot \sin(x) + (cx+d) \cdot \cos(x) + e$ for which I want to determine the roots. I know that for $ax \cdot \sin(x) + cx \cdot \cos(x)$ the roots are ...
0
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3answers
34 views

How to make a semicircle graph?

What is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks
0
votes
1answer
31 views

Find the sides of a right triangle formed by connecting two other right triangles from the center of their hypotenuse.

I have the following sketch of the problem: I need to find the values of $x$ and $y$ in the previous drawing. The hypotenuses of both black triangles are of equal length and the red triangle is a ...
7
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5answers
66 views

Proving that $\cos(\frac{\arctan(\frac{11}{2})}{3}) = \frac{2}{\sqrt{5}}$

I am trying to solve the cubic equation $x^3-15x-4=0$ using Cardano's formula. I already know that the solutions are $x=4$, $x= \sqrt{3}-2$ and $x= -\sqrt{3}-2$ and that using the formula in this ...
1
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1answer
42 views

Another way to calculate $\cos(x)$ and $\cosh (x)$

I usually don't see any expressions for approximate calculation of trigonometric functions aside from their Taylor series. But Taylor series work better for small angles, so in general they are not ...
0
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2answers
34 views

What trigonometric identity makes the method of triangulation work?

I've read the article on Wikipedia, but I don't get how to construct the relationships between sides and angles to reach a solution for the distance between two points. All the other sites I read ...
3
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1answer
29 views

Trigonometric integrals and limits

Show $$\lim_{N\to\infty}g_N(\theta_N)=2\int^\pi_0\frac{\sin x}{x}dx-\pi,$$ where $$g_N(\theta_N)=\int_0^{\theta_N}\frac{\sin[(N+1/2)x]}{\sin(x/2)}dx-\pi,$$ $$\theta_N=\frac{\pi}{N+1/2},$$ and ...
7
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2answers
116 views

Solving $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi ...
3
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1answer
63 views

Ratios of the major axes of ellipses inscribed in a triangle

I recently came across a question that went something like this: In a triangle with vertices (0,0), (6,0), (2,4) an ellipse is inscribed such that it has the largest area. Now it is dilated, that is ...
2
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2answers
35 views

The correct half angle formula?

It is well known that $$\cos(\frac x2)=\sqrt{\frac{1+\cos(x)}2}$$ And, we also know that $\cos(\frac x2)$ may be negative for some $x$ values. So that implies that: $$\cos(\frac ...
0
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0answers
40 views

Does $\lim\limits_{x\to0}x^2\csc\frac{1}{\sqrt[3]x}$ exist?

The original problem is computing the limit $$L=\lim_{x\to1}\frac{(x-1)^2}{\sin\frac{1}{\sqrt[3]{x-1}}}$$ for which I replaced $x-1$ with $x$. Is there something wrong with invoking the limit ...
7
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1answer
82 views

calculate $\frac{1}{\sin(x)} +\frac{1}{\cos(x)}$ if $\sin(x)+\cos(x)=\frac{7}{5}$

If $$\sin(x)+\cos(x) = \frac{7}{5}$$ Then what's the value of $$\frac{1}{\sin(x)} +\frac{1}{\cos(x)}\ \ \text{?}$$ Meaning the value of $\sin(x)$, $\ \cos(x)$ (the denominator) without using the ...
3
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6answers
72 views

How to proceed from $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$

To prove: $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$ My attempt at the solution: ...
0
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1answer
20 views

Drawing half a circle betweeen two arbitrary 2D points

So, I have two arbitrary points in a vector space and I'm trying to draw 180 degrees of a circle between them. The radius of the circle would be half of the distance between the two points and the ...
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1answer
35 views

solving polar simultaneous equations [on hold]

I need to solve the below polar equations. Question: Find all the points of intersections between the two polar curves s(θ)=(12θ,θ), r(θ)=(θ,12θ) Thanks
1
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2answers
21 views

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$ So $x\cot x = x\left( \frac{e^{ix}+e^{-ix}}{2}\cdot \frac{2i}{e^{ix}-e^{-ix}} \right) = \frac{ix(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}=\dots$ ...
0
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0answers
37 views

A trigonometric equation - Looking for closed form solutions.

I was wondering if the following equation has a solution \begin{equation} \frac{\sin\big[(N+1)\phi\big]}{\sin\big[N\phi\big]}=1+\frac{\alpha}{\cos\phi+\beta} \end{equation} where $\alpha$ and ...
2
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1answer
42 views

How can I find an output of this function's inverse without graphing?

How can I find $f^{-1}(5)$ where $$f(x)=\frac{27}{\pi}x + \sin x$$ algebraically? Thank you!!
1
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1answer
19 views

Calculate most parallel vector

Let's suppose I have a vector $\vec{a}$ and arbitrary many vectors (in my example two: $\vec{b}$ and $\vec{c}$), which all have a common starting point P. Now I want to find out which of these vectors ...
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1answer
46 views

General and principal solutions of Sinx + Sin3x + Sin5x = 0?

A solution that i found on net but cant figure out why we need to do Sinx + Sin5x Why does we take the sum of Sin x and Sin 5x ? Why cant we take the sum of Sin 3x and Sin 5x or Sin x and Sin 3x? ...
3
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3answers
56 views

Solution of the equation $\cot \theta = 2\cot 2\theta$

I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a ...
1
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1answer
33 views

If $A+B+C=π$, verify the given

If $A+B+C=π$, prove that $$\cos A \sin B \sin C + \cos B \sin C \sin A + \cos C \sin A \sin B=1+\cos A \cos B\cos C$$ ATTEMPT: Here, $$A+B+C=π$$ Now, \begin{align*} \text{L.H.S} &= \cos A \sin B ...
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4answers
61 views

Hint: $\sin^2(6x)-\sin^2(4x) = \sin(2x)\sin(10x)$ [duplicate]

I need a hint to solve prove $\sin^2(6x)-\sin^2(4x) = \sin(2x)\sin(10x)$ I tried several solutions, including taking $(\sin(6x)+\sin(4x))(\sin(6x)-\sin(4x))$ but every time I ended up with a ...
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1answer
28 views

Find both the diagonals with area and side given [on hold]

Side of rhombus $= 65$ cm Area of rhombus $= 1024$ cm$^2$ Find the diagonals of the rhombus.
3
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0answers
55 views

Want to know what's wrong?

I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ...
0
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0answers
36 views

An inequality with the sinus in a triangle.

I have solved this problem in a way, rather "inspired". I would like to have a solution found an easier way but I was unable so far. Let $A,B,C$ the angles of a triangle $\triangle {ABC}$; prove ...
4
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1answer
94 views

Is $\arctan2$ irrational? [duplicate]

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
1
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1answer
38 views

the maxima of given function

What's the maxima of $$2^{\sin(x)}+2^{\cos(x)}$$ I found max by taking logs and then differentiating and equating to $0$ at $x=45°$ so the answer is $2^{\frac{\sqrt{2}+1}{\sqrt{2}}}$ am I right or I ...
2
votes
2answers
39 views

Find the value of $ sin(2\theta)$ when $cot(\theta) + tan(\theta) = 2.5 $

I have an homework question that goes like: $cot(\theta) + tan(\theta) = 2.5 $ is valid on some angles $\theta$ at section $0 < \theta < \pi/2$. Find the value of $sin(2\theta)$. (There is no ...
0
votes
1answer
11 views

Having trouble drawing an arc using atan2 and specified range of arc.

I'm writing code. My arc invariant is: $$ \theta_0, \theta_1 \in [-2\pi, 2\pi], \\ \theta_1 - \theta_0 \leq 2\pi $$ where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle. ...
3
votes
1answer
33 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
0
votes
0answers
10 views

Find largest value of the constants that makes the equality true

$$sin(h^2) = a + O(h^b)$$ Find the largest value of a and b such that the equality holds. I tried to use the truncated Taylor series expansion of $sin(h^2)$, but the derivatives of the function are to ...
1
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0answers
32 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...