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

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238 views

Having trouble with trig inequalities

If someone could help with either of these problems that would be awesome! $(\tan x)^2 \leq |1 - 2(\cot x)^2|$ $x^{\sin(x-a)}>1$ where ($0< x < \frac{\pi}{2}$, $a>0$)
3
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2answers
234 views

How to simplify trigonometric inequality?

$| 3 ^ { \tan ( \pi x ) } - 3 ^ { 1 - \tan ( \pi x ) } | \geq 2$
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1answer
1k views

Positioning three circles, all of them touching each other

There are three circles, all of them touching each other. The bottom two circles are laying on an imaginary floor, such that they touch the line g=-r as well. Given are all three radii, r1 (A), r2 ...
0
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2answers
112 views

Vector constrained to an angle and length?

I would like to like to be able to find the point p3 given point p1 and p2. The length between point p1 and p2 is set to L and the length between point p2 and p3 is set to L. The angle between point ...
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3answers
1k views

angle=arctan(dy/dx), what happens when dx=0

here is a fomula: angle=arctan(dy/dx) I can find an angle with my calculator for any value except dx=0, my question is: Is there no angle or , there is something that says when dx==0 the angle=?... ...
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6answers
1k views

Pointwise Convergence of $\sum \frac{\sin(\sqrt{n}x)}{n}$

I am having trouble in proving the pointwise convergence of $$ g(x)=\sum_{n=1}^\infty \frac{\sin(\sqrt{n}x)}{n}$$ for all real numbers $x$ using elementary methods (e.g. integral test, Weierstrass ...
3
votes
1answer
239 views

How can I determine a number is irrational?

I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be ...
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3answers
2k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
6
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1answer
799 views

Divergence of $\sum\frac{\cos(\sqrt{n}x)}{\sqrt{n}}$

I have difficulties in showing the series $f(x)=\sum_{n=1}^\infty \frac{\cos(\sqrt{n}x)}{\sqrt{n}}$ is divergent at every real numbers $x$. However I cannot find any elementary methods to do this. ...
0
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2answers
497 views

How to solve trigonometric equation $\sin(x)+x\cdot \cos(x)=0$?

I'm facing the problem of solving $$\sin(x)+x \cdot \cos(x)=0$$ using $$\tan(x)=\sin(x)/\cos(x)$$ I end at $$x+\tan(x)=0$$ on the other hand, I also tried $\cos(x)= \pm ...
6
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3answers
747 views

Solve $\sin(5A) + \cos(5A)\sin(A) - \cos(3A) = 0$

How do you solve this equation for A: $~~\sin(5A) + \cos(5A)\sin(A) - \cos(3A) = 0$ I've tried expanding it many times, but I can't seem to be able to reduce it to a format I can work with. Is there ...
8
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1answer
483 views

Solving $\displaystyle \cos(x-\alpha)\cos(x-\beta) = \cos{\alpha}\cos{\beta}+\sin^2{x}$

Solve $\displaystyle \cos(x-\alpha)\cos(x-\beta) = \cos{\alpha}\cos{\beta}+\sin^2{x}$. My attempt: $\displaystyle \cos(x-\alpha)\cos(x-\beta) = \cos{\alpha}\cos{\beta}+\sin^2{x} \Rightarrow ...
3
votes
2answers
161 views

Determining Angle

If we are given two sides say $a$,$b$ and an angle $X$,How can we determine whether this angle $X$ is opposite angle to $a$ or $b$ (i.e $A$,$B$) or the third included angle $C$ ?
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5answers
3k views

Solve $\sin x = 1 - x$

How would you be solve sin x = 1 - x, without drawing the graph and manually measuring the point of intersection?
5
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3answers
167 views

trigonometric inequality

please help me establish(etablir): $\forall n\in \mathbb{N}-\left\{ 0,\left. 1 \right\} \right.$ , $x\in \mathbb{R}-\left\{ \pi \mathbb{Z} \right\}$ , $\left| \sin \left( nx \right) ...
2
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1answer
150 views

Simplify Trig Expressions

I need to simplify $\sin^3(x)+\cos^2(x)\sin(x)$: First thing I noticed was the pythagorean identity. $$\sin(x)\sin^2(x)+\cos^2(x)\sin(x) \rightarrow \sin(x)(1)\sin(x)\rightarrow\sin^2(x),$$ but this ...
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1answer
2k views

Simplifying $\sqrt{1+cos^2(x)}$ for various values of x

I'm being asked to find the arc length of $y=sin x$ for [0, $\frac{pi}{2}$] using $M_8$. I've determined that $y\prime^2=cos^2x$. So, using the formula for arc length, I get $\sqrt{1+cos^2x}$ as my ...
5
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3answers
831 views

Finding $ \csc \theta $ given $ \cot \theta $

I have the following problem: If $ \cot{C} = \frac{\sqrt{3}}{7} $, find $ \csc{C} $ From my trig identities, I know that $ \cot{\theta} = \frac{1}{\tan{\theta}} $, and $ \csc{\theta} = ...
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3answers
3k views

Chain Rule applied to Trig Functions

Given $f(x)= \sin(\pi x)^{2}$, find the derivative. Using the chain rule my work is as follows: $(\sin(\pi x)^2)'$ becomes $$2 \sin(\pi x) \cdot \frac{d}{dx}(\sin(\pi x)$$ The derivative of sin ...
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3answers
3k views

Finding a point on the unit circle; more specifically, what quadrant it is in

In my Trig class we have begun working on graphing the trig functions and working with radians and I'm trying to wrap my head around them. At the moment I'm having trouble understanding radian ...
4
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3answers
586 views

Trig identity proof help

I'm trying to prove that $$ \frac{\cos(A)}{1-\tan(A)} + \frac{\sin(A)}{1-\cot(A)} = \sin(A) + \cos(A)$$ Can someone help me to get started? I've done other proofs but this one has me stumped! Just a ...
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3answers
738 views

$\sin(2\pi nx)$ does not converge for $x \in (0,1/2)$

How to show that $\sin(2 \pi nx)$ does not converge as n goes to infinity? $x \in (0,1/2)$
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1answer
3k views

2D Rotation Around Point

D. My first post here ::- >. I got a rather simple question. But please, allow me to introduce myself a bit first. I think it's polite for a first post ::- D. I'm a game developer (free Flash games) ...
5
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1answer
280 views

Simplifying $\sum 2^k \tan(2^k x)$

Simplify $\sum\limits_{k = 0}^n {{2^k}\tan ({2^k}x)}$ which $k \in \{ 0,1,...,n + 1\} ,{2^k}x \notin \{ 0,\frac{\pi }{2}\}$
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votes
1answer
224 views

Given a point and a set of triangles, what's would be a fast way to find which triangle the point belongs to?

I'm trying to do a piece-wise affine transform in Python. I have one image with a set of points hand marked and another set of points where I wish to "move" my current points and the texture between. ...
4
votes
4answers
2k views

Trying to derive an inverse trigonometric function

I'd like to know how to derive these functions (I know the answers, I want to know how to get there) \begin{align*} f(x) &= \arcsin\left(\frac{x}{3}\right)\\ f(x) &= \arccos(2x+1)\\ f(x) ...
2
votes
4answers
13k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
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1answer
171 views

What conic curve is the graph of y=sin(x) from 0 to pi?

Certainly is not a circle; it looks like an ellipse but I don't think it is. Thanks
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3answers
804 views

Simultaneous equations, trig functions and the existence of solutions

Came across this conundrum while going over the proof that $$A \cdot \sin(bx) + B \cdot \cos(bx) = C \cdot \sin(bx + k)$$ for some numbers $C$ and $k$. ($A$, $B$ and $b$ are known.) The usual method ...
2
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2answers
290 views

Solving short trigo equation with sine - need some help!

From the relation $M=E-\epsilon\cdot\sin(E)$, I need to find the value of E, knowing the two other parameters. How should I go about this? This is part of a computation which will be done quite a ...
13
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4answers
944 views

Finding $\int_0^{\pi/2} \sin x\,dx$

I'm interested in why $$\int_0^{\pi/2} \sin x\,dx = 1.$$ I know how to do the integral the conventional way but am more interested in what makes radians special for this problem. If we instead compute ...
2
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2answers
554 views

Trig Question - $\arcsin(\sqrt{2}/2)$ and arc trig functions in general

I know $\arcsin(\sqrt{2}/2)$ is equal to $\pi/4$. However I don't understand why, I've done some searching on google about arc trig functions and I haven't found any webpages that explain it very ...
2
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2answers
253 views

What is the simplification of $\sin^2 x/(1+ \sin^2 x +\sin^4 x +\sin^6 x + \cdots)$?

What is the simplification of $\sin^2 x/(1+ \sin^2 x +\sin^4 x +\sin^6 x + \cdots)$?
9
votes
2answers
529 views

New size of a rotated-then-cropped rectangle

Imagine a rectangle (x1 by y1) always has to be drawn with horizontal and vertical lines (so it can't have lines at 45 degrees). If the rectangle is rotated by angle θ, it needs to have a rectangle ...
1
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2answers
203 views

Simplifying a Trigonometric Expression

I have to prove that: $$x \sec x - \ln |\sec x + \tan x| + C$$ is the indefinite integral of: $$x \sec x \tan x $$ by taking the derivative. I've got far enough to get: $$x\sec x\tan x + \sec x ...
3
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1answer
242 views

Graph for $f(x)=\sin x\cos x$

Okay, so in my math homework I'm supposed to draw a graph of the following function: $$f(x)=\sin x \cos x.$$ I have the solution in the textbook, but I just can't figure out how they got to that. So, ...
10
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5answers
2k views

Solve $\cos(\theta) + \sin(\theta) = x$ for known $x$, unknown $\theta$?

After looking at the list of trigonometric identities, I can't seem to find a way to solve this. Is it solvable? $$\cos(\theta) + \sin(\theta) = x.$$ What if I added another equation to the problem: ...
2
votes
1answer
357 views

How to solve algebraically the equation $x = \frac{1}{2}\cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$

How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ? The iterative solution seems to be 1.417. Can anybody suggest an algebraic ...
2
votes
3answers
486 views

Why are there two possible triangles when given SAS?

I gave my trigonometry students the following example: Solve $\triangle ABC\ $ , where AC=0.923, AB=.387, and $\measuredangle A\ = 43.33^\circ\ $. First I found BC using the law of cosines, then I ...
3
votes
3answers
282 views

Trigonometric equality

I would like to know, how do you simplify this: $$\cos x\sin(x+y) + \sin x\cos(x+y)$$ to this: $$\sin(2x+y).$$ Wolfram alpha says so, but how does human being do so? :)
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2answers
115 views

Radius of a hypercube at a given angle

For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this: Given ...
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3answers
1k views

Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$

Again: $$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$ Also the one for $\sin$: $$\int e^{\alpha x}\sin(\beta x) ...
2
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2answers
156 views

Trigonometry Expression

Is $(\sin \phi)^2$ is equal to $\sin^2\phi$? Can any one tell what is the ans for the below expression $\sin^260$ + $\cos^260$ + $\tan^245$ + $\sec^260$ - $\csc^260$
3
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2answers
135 views

Help with using some trig identities

Need some help with the steps in converting the derivatives of the following functions. derivative of $\cos(\tan(x))$ to $\frac{-\sin(\tan (x))}{\cos^2(x)}$ I can get $-\sec^2(x) \cdot ...
6
votes
5answers
998 views

If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$ then $\tan x + \cot x=?$

Hello :) I hit a problem. If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$, then how much is $\tan x + \cot x$?
2
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2answers
624 views

How can I solve for a single variable which occurs in multiple trigonometric functions in an equation?

This is a pretty dumb question, but it's been a while since I had to do math like this and it's escaping me at the moment (actually, I'm not sure I ever knew how to do this. I remember the basic ...
2
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4answers
776 views

What is the limit as $x\to\infty$ of $\cos x$?

What is the limit as $x\to\infty$ of $\cos x$? Thanks in advance.
4
votes
5answers
624 views

Simpler solution to this geometry/trig problem?

i had a geometry/trignometry problem come up at work today, and i've been out of school too long: i've lost my tools. i'm starting with a rectangle of known width (...
3
votes
1answer
113 views

Proving equation at zero?

I have an equation $$x = \csc(\theta) - \cot(\theta).$$ As $\theta$ approaches zero, $x$ approaches zero. However, trying to solve the equation at zero yields an undefined result. How do I rewrite ...
10
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2answers
3k views

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

How can we sum up $\sin$ and $\cos$ series when the angles are in A.P (arithmetic progression) ?For example here is the sum of $\cos$ series: $$\large \sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n ...