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

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3
votes
2answers
61 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, ...
7
votes
1answer
52 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 ...
0
votes
3answers
25 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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
24 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} ...
0
votes
1answer
14 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 ...
2
votes
1answer
28 views

Geometrically interpreting complex numbers.

Prove that $|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$ ...
0
votes
2answers
15 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 $-\frac{\sqrt{3}}{2}$?
0
votes
4answers
52 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 ...
4
votes
3answers
516 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 + ...
1
vote
3answers
30 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
votes
2answers
64 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
vote
1answer
23 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
votes
2answers
43 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 ?
0
votes
2answers
40 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
votes
0answers
14 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
vote
4answers
64 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
votes
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
vote
2answers
50 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 ...
-3
votes
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
votes
1answer
34 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
vote
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
votes
4answers
68 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
votes
1answer
57 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
votes
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
vote
2answers
36 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
vote
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 ...
2
votes
2answers
49 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
votes
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
32 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
votes
5answers
67 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
vote
1answer
45 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
votes
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
votes
1answer
31 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
votes
2answers
117 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
votes
1answer
66 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
votes
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 ...
0
votes
0answers
41 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
votes
1answer
84 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
votes
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
votes
1answer
21 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 ...
-3
votes
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
vote
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
votes
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
votes
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
vote
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 ...
-1
votes
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
votes
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
vote
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 ...
0
votes
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 ...