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

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1answer
236 views

Line Rotation Problems in Trigonometry

I have 2 lines that i draw like this: ...
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1answer
173 views

Calculate angle of a line at the point it intersects a circle

Given a line with end points $(x_1,y_1)$ $(x_2,y_2)$ and a circle centered at $(x_1,y_1)$ how do I calculate the angle of the line (in degrees) as it relates to the circle? If that doesn't make sense ...
7
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2answers
244 views

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities?

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities? In functions like $\tan^{-1}(a \tan(x))$, the angle wraps and the result is discontinuous. Is it legitimate to redefine the ...
4
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2answers
433 views

Proving $\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1$

I was stumped by another past-year question: In $\triangle ABC$, prove that $$\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1.$$ Here's what I have done so far: I tried to replace $C$, using ...
5
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1answer
99 views

Some trigonometric formula

How to prove that $1+2(\cos a)(\cos b)(\cos c)-\cos^2 a-\cos^2 b-\cos^2 c=4 (\sin p)(\sin q) (\sin r)(\sin s)$, where $p=\frac{1}{2}(-a+b+c)$, $q=\frac{1}{2}(a-b+c)$, $r=\frac{1}{2}(a+b-c)$, ...
3
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2answers
238 views

Weird derivative of $\tan^{-1} x$

I've seen this in Stewart calculus book: $$\frac{\mathrm d \tan^{-1} x}{\mathrm dx} = \frac1{1+x^2}$$ But how do I get it? If I do it myself, $$\frac{\mathrm d \tan^{-1} x}{\mathrm dx} = ...
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1answer
7k views

How to find the Period and Phase angle?

I'm currently brushing up my trig and found these two problems. I'm totally clueless on how to start. Please help. Find the period , amplitude , and phase angle, and use these to sketch a) ...
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1answer
156 views

2 solutions for, solve $\cos x = -1/2$? Answer sheet displays only one, does this mean there is only one?

$\cos x = -1/2$ can occur in quadrants 2 or 3, that gives it 2 answers, however the answer sheet only shows one. Does this mean im doing something completely wrong, or are they just not showing the ...
6
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1answer
343 views

Proving that if $\cos{x} = \cos{y}$ and $\sin{x} = \sin{y}$ then $x-y = 2\pi n$ for some $n\in \mathbb{Z}$

I was solving some exercises in complex analysis in preparation for a qualifying exam, and I came across a problem which asked me to prove that if $x, y \in \mathbb{R}$ then $$ e^{ix} = ...
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2answers
204 views

Solving $\sin{(\cdot)}, \cos{(\cdot)}, \tan{(\cdot)}, \cot{(\cdot)} \dots$ without a calculator.

For example solve $\cos{\left(\frac{5\pi}{4}\right)}$ without a calculator or solve $\cos{(x)} = -\frac{1}{2}$. I remember vaguely that the method involves referring to a triangle, but im not sure. ...
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3answers
432 views

Are there any natural occurrences of taking a trig function of a trig function?

In devising challenging exercises for my students, I am tempted to give them something like $\cos(3\sin(4))$, but then I get to wondering whether such a calculation would ever be encountered in ...
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2answers
134 views

Finding: $\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$

How to find: $$\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$$ I know that $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}$ but don't know how to apply it here.
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2answers
5k views

Deriving 2D Coordinate Rotation Formula

I'm trying to write out the steps in code for deriving the 2D coordinate rotation formula so I can understand it. ...
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0answers
77 views

Counteract preceding rotations

In a situation where I have two axis adjacent back to back (let's say a robotic arm) I can sometimes perform two rotations ($R_1, R_2$) such that the resulting position and direction is unchanged. ...
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4answers
1k views

Best way to find the Coordinates of a Point on a Line-Segment a specified Distance Away from another Point

I have 4 points: $Q, R, S, T$. I know the following Coordinates for $R$, $T$, and $S$; Length of $\overline{RQ}$ That segment $\overline{RT} < \overline{RQ} < \overline{RS}$; I need to ...
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2answers
266 views

Simplify using the Tangent Difference identity

I solved a problem to the point that I know the answer is $$\frac{2nr}{\tan\left( \frac{(n-2)\pi}{2n} \right)}$$ The question tells me that the answer is going to be ...
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0answers
113 views

Solving a linear trigonometric equation

Let $n$ be a natural number. For $a_i,\omega_i,\varphi_i \in \mathbb{R}$ how can one find solutions $x \in \mathbb{R}$ for the equation: $$\sum_{i=1}^n a_i \cos( \omega_i \cdot (x-\varphi_i)) = 0$$
3
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1answer
691 views

Integral of exponential function with trigonometric identities

I need help in solving the following definite integral. I could not find any example like this $$\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; ...
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3answers
1k views

Canonical to Parametric, Ellipse Equation

I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things. $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ...
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4answers
1k views

Resizing a rectangle to always fit into its unrotated space

(For those coming here looking for answers to rectangle problems it may help to see the related (and solved) question: Given a width, height and angle of a rectangle, and an allowed final size, ...
6
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1answer
219 views

Trigonometric system

I would like to solve: $ x +y+z=\frac{11\pi}{6} $ $ \sin(x)+\sin(y)+\sin(z)= \frac{\sqrt{3}}{2} $ $ \cos(x)+\cos(y)+\cos(z)=\frac{1}{2} $ After eliminating $ z $ I get: $ ...
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2answers
2k views

Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area

In other words, if I had a rectangle of $10\times 10$ and an angle of $45$, and the allowed area was $100\times 100$, the rectangle would be about $70\times 70$. The allowed area is $100\times 100$ ...
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3answers
630 views

Limits with trig functions

I've been slacking off in calculus and I honestly don't know what to do with this problem. I got that it's in the indeterminate form, but I have no clue where to go from there. $$ \lim_{x \to 0} ...
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3answers
1k views

Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}\ ...
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1answer
359 views

I have equations for getting x,y,z given latitude, longitude, and altitude. How do I reverse them?

I am using equations that look like the following to get x, y, and z given latitude, longitude, and altitude. ...
2
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1answer
2k views

Find angle between two lines

I have two lines. I have each X,Y. I want to find the angle between them (please mark if your method returns radians or degrees) Aka a function F[x,y,x2,y2,Cx,Cy] that'll return their angle (at ...
7
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1answer
656 views

Trying to get the infinite product for $\sin x$

I start with the fact that the zeros of $\sin x$ are $ n\pi$, $n\in\mathbb{Z}$. Therefore, it should be possible to express it as an infinite product: $$\sin x = x ...
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2answers
155 views

Integration - Primitives - Antiderivatives

Please help to calculate: $$\int\sqrt {{r}^{2}-{x}^{2}}{dx},\quad x\in[0,r]$$ Do any method of trigonometric substitution? Thanks.
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1answer
298 views

Trigonometry, some true or false tasks about cosine-rule and sine-rule

I don't understand why the answer is as it says in the book. Let me write out the task first. A If the cosine-rule can be used to find an angle in a triangle, it is only one angle that fits this ...
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2answers
116 views

Prove that $\vert\sin(x)\vert + \vert\sin(x-1)\vert \ge \sin(1)$

While looking into the convergence of the series $\sum_{n=1}^{\infty}\frac{\sin(n)}{n}$ I stumbled into the inequality $\vert\sin(n)\vert + \vert\sin(n-1)\vert \ge \sin(1)$ for all $n\in\mathbb{R}$. ...
7
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3answers
672 views

What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$

Here is a nice graph representing the solution: wolframalpha. I wish to draw such a graph myself but don't have any idea which methods exist and which of them are more appropriate for equations of ...
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1answer
846 views

Find the maximum area possible of equilateral triangle that inside the given square

How can I find the maximum area possible of equilateral triangle that inside a square whose sides have length a. And how does that triangle look like? Can we construct it (with compass and ...
10
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3answers
1k views

Show that $A_n=\sum\limits_{k=1}^n \sin k $ is bounded?

Let $A_n=\sum\limits_{k=1}^n \sin k $ , show that there exists $M>0$ , $|A_n|<M $ for every $n$ .
11
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1answer
290 views

Trigonometric identity

It is well known (?) that if $\alpha+\beta+\gamma=\pi$ then $4\sin\alpha\sin\beta\sin\gamma = \sin(2\alpha)+\sin(2\beta)+\sin(2\gamma)$ (I think I've seen it in some late-19th-century books, and I ...
3
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2answers
210 views

Complex number + trigo : $-1 + \tan(3)i$ , find modulus and argument

I have $-1 + \tan(3)i$ and must find its modulus and its argument. I tried to solve it by myself for hours, and then I looked at the answer, but I am still confused with a part of the solution. Here ...
2
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2answers
57 views

Movement depending on angle value

I have an object and an angle value (clockwise) in my game. Depending on the angle, the object will move toward a certain direction. If the angle is 0, the object will move (0,1) per frame (x,y) ...
3
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1answer
4k views

Calculations of angles between bonds in CH₄ (Methane) molecule

In my high school chemistry class, we talked about the angles between bonds in molecules. One that caught my attention was the CH₄ molecule. I asked my teacher how to calculate this result, he said ...
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4answers
4k views

What is the maximum value of this trigonometric expression

What is the maximum value of the expression $1/(\sin^2 \theta + 3\sin\theta \cos\theta+ 5\cos^2 \theta$). I tried reducing the expression to $1/(1 + 3\sin\theta$ $\cos\theta + 4\cos^2 \theta)$. How ...
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1answer
368 views

Trigonometry, height of mountain based on angle difference?

The text translated: From a point A on a water we observe a antenna on a mountain. The antenna is 23m high. Find the height of the mountain. I believe the solution is in the differences between ...
3
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1answer
184 views

trigonometric system

In order to show that $ e^{ix}+e^{iy}+e^{iz}=0 \Longrightarrow e^{2ix}+e^{2iy}+e^{2iz}=0 $, I want to prove that $ \cos x+\cos y+\cos z=0 $ and $ \sin x+\sin y+\sin z=0 \Longrightarrow \cos 2x+\cos ...
2
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1answer
119 views

Is there a way to solve for $x$ in $\cos(ax)/\cos(bx)=c$?

Is there a way to solve for x in $\dfrac{\cos(ax)}{\cos(bx)} = c$? This is similar to the question on $\dfrac{\cos^{-1}(ax)}{\cos^{-1}(bx)} = c$.
2
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1answer
89 views

How can I calculate $a$, $f$, $x_0$ and $y_0$ in $y = a \cos(f(x - x_0)) + y_0$ given four arbitrary points?

How can I calculate $a$, $f$, $x_0$ and $y_0$ in $y = a \cos(f(x - x_0)) + y_0$, given four arbitrary points $(x_1, y_1)$, $(x_2, y_2)$, … that the graph must go though? The complexity of ...
2
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1answer
119 views

Is there a way to solve for $x$ in $\cos^{-1}(ax) / \cos^{-1}(bx) = c$?

Is there a way to solve for $x$ in $\dfrac{\cos^{-1}(ax)}{\cos^{-1}(bx)} = c$? I guess it comes down to, are there any sine multiplication formulas I don't know about? The motivation for this is ...
2
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1answer
198 views

Cosine Trigonometry Question

Find the radian measure of $\theta$ if $0 \leq \theta \leq 2\pi$ and $$\cos(\theta)(2\cos(\theta)-1) = 0.$$ I'm very new to this topic, so what I did was to take the inverse of $\cos$ from both sides, ...
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1answer
261 views

Trigonometry, Pythagorean theorem, how to apply it in this task?

Let me first write out the task. A square property ABCD, Angle A = B 90 degrees. AB = 39m, BC = 32m, and the diagonal BD is 55m. How long would a fence be to run around the property? I drew it up ...
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3answers
260 views

Algebra in trigonometry, algebraic proof?

The picture says it all. "Vis at" means "show that". My first thought was that h is 2x, which is not correct. Maybe the formulas for area size is useful? EDIT: (To make the question less dependent ...
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1answer
198 views

A boundary value problem over an infinite interval

This is the edited version of the original problem, hopefully presented in a clearer manner. (I have also renamed this post with a more befitting title) Problem: $$y'(x) = ...
5
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2answers
2k views

Where are the zeros of the complex sine and cosine?

Do sin(z) and cos(z) have any zeroes where the imaginary part of z is non-zero? How could I prove that (or show that it's reasonable)?
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3answers
2k views

How do I calculate the phase shift between sine and cosine?

I know that $\sin(\alpha + x)=\cos(\alpha)$. How do I find $x$ ? I'd start by using the angle sum identity for sine: $\cos(\alpha)*\sin(x)+\sin(\alpha)*\cos(x)=\cos(\alpha)$ I had some ideas about ...
6
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4answers
1k views

Infinite series expansion of $\sin (x)$

Are there any other ways to demonstrate that $$\sin(x)=\sum_{k=0}^{\infty}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$$ without using the definition of Taylor series of complex exponentials, and similarly for ...