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

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Elementary 3D geometry

This is surely trivial, but my old brain can't remember how to do it. Assume a plane. A second plane intersects, forming line $AB$. The angle of intersection is $\theta$. A third plane intersects, ...
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20 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
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1answer
31 views

Rotating a sphere

I'm trying to rotate a sphere, and I'm having a bit of a problem calculating the angle to rotate it by. I wonder if anyone can help me? On my sphere I've marked three points. If the centre of the ...
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2answers
42 views

Calculating trigonometric function values mentally

This may sound dumb, but does such a way exist to mentally (and quickly) determine the values of trigonometric functions such as $\sin(47^\circ)$ and so forth--quickly being a mere matter of seconds? ...
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23 views

Find $\angle B$ if $AD=\frac{abc}{b^2-c^2}$

If AD is median and $AD=\frac{abc}{b^2-c^2}$ $[b>c]$ and $\angle C=23^{\circ} $. Find $\angle B$ Is this information sufficient to find $\angle B$? I tried using sine rule in triangle $ADC$ and ...
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0answers
14 views

Length and width of shadow of rectangular plane

A book that I've read shows how to find the area of the shadow cast by a sphere and ellipsoid. The spherical shadow makes sense; its simply the area of a circle (which would be the sphere's shadow) ...
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2answers
25 views

Can hyperbolic sine and cosine be combined into a single function of shifted argument?

For trigonometric functions we have a nice identity: $$A\cos x+B\sin x=\sqrt{A^2+B^2}\sin(x+\operatorname{atan2}(A,B)).\tag1$$ At the core of it is the well-known identity of ...
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14 views

Find the coordinate of a tetrahedron knowing all side lengths

I know the coordinates of three points $(x_1, y_1, z_1)$, $(x2_, y_2, z_2)$ and $(x_3, y_3, z_3)$. Now I need to localize a fourth point $(x_4, y_4, z_4)$. I therefore get all the distances $d_{41}$, ...
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28 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\sum_{r=0}^{N-1}\frac{\sin^2(\frac{\pi(Ne-e-r+n))}{N})}{\sin^2(\frac{\pi(r-n+e))}{N})}$$ and I want to show analytically that for small $e$ ($e<0.2$) and large ...
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1answer
47 views

How do I prove that sine is dependent only on the angle? [on hold]

I did the following: Taking one triangle and writing the pythagorean formula to it: $a^2+b^2=c^2$ and hence: $$\frac{o}{h}=\frac{\pm \sqrt{c^2-a^2}}{\pm \sqrt{a^2+b^2}}$$ I took another ...
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1answer
57 views

Height and Distance problems

A ladder rests against a wall at an angle $\alpha$ to the horizontal. When its foot is pulled away from the wall through a distance $a$, it slides a distance $b$ down the wall and makes an angle ...
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2answers
32 views

Simplify the expression and leave answer in terms of $\sin x$ and/or $\cos x$

$1-\sin^2 x = \cos^2 x$. However, $1-\sin^2 x$ can also be factored using the difference of two squares. I am stuck on whether $1- \sin^2 x$ should turn into $\cos^2 x$ or be factored by using the ...
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3answers
97 views

If $\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0$,

If $$\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0,$$ prove that $$\cos 3A+\cos 3B+\cos 3C=3\cos(A+B+C).$$ My solution: From the given, $$\cos^3A+\cos^3B+\cos^3C=3\cos A\cos B\cos C$$ Now, ...
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1answer
85 views

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function ...
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3answers
34 views

Inverse Trig and infinite values (arccos)

I understand that trig ratios can have infinite values for the same value of $x$ $ \cos(x) $ for example. Since $ \cos(x) $ shows the relationship between two sides of a triangle and that ratio can ...
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1answer
24 views

Precision of Manual Vector Addition

I learned the fundamentals of vectors and basic (e.g. addition, dot product) vector operations in a Trigonometry course, and they're being reintroduced in the Physics I course I just began. My ...
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1answer
25 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
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1answer
15 views

Principal solution of $ \sin(x)=1$ from $-2 \pi$ to $2 \pi$

Solve $\sin(x)=1$ for values of $x$ where $-2\pi\le x\le 2\pi$ Now, I know that $sin(\pi/2$)=$1$ in 1st quadrant and by using $sin(\pi-x)=sin(x) $ I still have $\pi/2$ and by using ...
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1answer
30 views

Geometrically interpreting complex numbers.

Prove that $|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$ ...
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2answers
17 views

How do I find remaining trigonometic function if $\cos(x)$ is negative?

What is the mistake in my method because the correct value of $\sin(x)$ is $-\frac{\sqrt{3}}{2}$?
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4answers
76 views

How to find $\lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2}$

By plotting $\dfrac{1-\cos x \sqrt{\cos 2x}}{x^2}$, we find that in sufficiently small domain near $x = 0$, $f(x)\to 0$ as $x\to 0$. So the limit seems to be $0$. Now I tried to evaluate it using ...
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3answers
524 views

Solving a trigonometric equation

I'm solving this equation: $$\sin(3x) = 0$$ The angle is equal to 0, therefore: $$3x=0+2k\pi \space\vee\space3x= (\pi-0)+2k\pi$$ $$x = \frac {2}{3}k\pi \space \vee \space x = \frac {\pi + ...
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3answers
32 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 ...
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2answers
66 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{ ...
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1answer
27 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 ...
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2answers
47 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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41 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 $|z|\gt \frac{3}{4}$.
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17 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 ...
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4answers
65 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 $$ ...
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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 ...
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2answers
56 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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29 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? ...
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1answer
36 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 ...
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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 ...
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4answers
72 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\\ & = ...
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1answer
62 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 ...
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19 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 ...
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2answers
38 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 ...
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1answer
48 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 ...
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2answers
51 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 ...
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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
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1answer
33 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 ...
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5answers
71 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 ...
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1answer
47 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 ...
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2answers
37 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
32 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
121 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
68 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
36 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 ...
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0answers
44 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 ...