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
74 views

How zeros of second derivative map to equilibrium value

This question arose from plotting some functions in MATLAB and GeoGebra. Assume we have a function of the form $$f(x)=A \sin(P_n(x))+d \qquad \text{or}\qquad f(x)=A \cos(P_n(x))+d$$ where $$P_n(x) ...
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1answer
190 views

Looking for good visual aids for sin cos tan .

Looking for good pictures ( or videos ) with explanations for visualizing sin cos and tan. Any relation to 3D Graphics is a bonus.
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4answers
651 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
2
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1answer
865 views

How to simultaneously solving trig equations?

I was doing some math work, and have to solve two trig equations simultaneously, but have no idea how to approach this, can anyone help, just need to be pointed in the right direction. I have to ...
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1answer
210 views

Expressing $\int \tan^n x\,dx$ with a sum

I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that $$\begin{align} \int \tan^n x\,dx ...
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1answer
470 views

Show that $2θ + 2\sinθ - 1 = \pi/3.$

Need help with this question: The diagram shows that the cross section ABCD of a glass prism. AD = BC = 4cm and both are at right angles to DC. AB is the arc of a circle, centre O and radius 6cm. ...
2
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0answers
249 views

Product of Sine: $\prod_{i=1}^n\sin x_i=k$

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin ...
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1answer
554 views

$\cos^n x-\sin^n x=1$

For $0 < x < 2\pi$ and positive even $n$, the only solution for $\cos^n x-\sin^n x=1$ is $\pi$. The argument is simple as $0\le\cos^n x, \sin^n x\le1$ and hence $\cos^n x-\sin^n x=1$ iff $\cos^n ...
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4answers
134 views

Lost in Trig Identities

I've been working on this problem on and off for a couple hours and have not been able to find out how to go about it even after looking through the other questions and some google searching. Prove ...
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2answers
150 views

Derivatives of trig functions

How can I prove that $\frac{d}{dx} (\csc x)= -\csc x \cot x$? Specifically, how does one see the step $\cos x/\sin x = \cot x$?
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1answer
91 views

Why $\sqrt{\sin^2 x}<0.5$ can be transformed in $|\sin x|<0.5$?

Why $\sqrt{\sin^2 x}<0.5$ can be transformed in $|\sin x|<0.5$. Then $|\sin x|<0.5$ can be transformed in $-0.5<\sin x<0.5$? What is the proof of the inequality?
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1answer
135 views

A curious identity on sums of secants

I was working on proving a variant of Markov's inequality, and in doing so I managed to come across an interesting (conjectured) identity for any $n\in\mathbb{N}$: $$\sum_{m=0}^{n-1} ...
3
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2answers
857 views

How many distinct real root does the equation $x^{4}-x^{3}\cdot\sin(x)-x^{2}\cdot\cos(x)=0 $ have?

How many distinct real root does the equation $x^{4}-x^{3} \cdot \sin(x)-x^{2} \cdot \cos(x)=0 $ have? Is there any quick solution(less than 2 minutes)?
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5answers
409 views

Prove the trigonometric identity $(35)$

Prove that \begin{equation} \prod_{k=1}^{\lfloor (n-1)/2 \rfloor}\tan \left(\frac{k \pi}{n}\right)= \left\{ \begin{aligned} \sqrt{n} \space \space \text{for $n$ odd}\\ \\ ...
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2answers
326 views

What is the limit of this function as $n$ tends to infinity?

$\lim_{n\rightarrow\infty}\sqrt{n}\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$ I'm having a lot of trouble figuring it out. My first step is always to convert $\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$ to ...
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3answers
360 views

Clarification on Trigonometric Notation

Consider the following terms: $cos5x$ and $sin^2x$ Are these terms equivalent to: $5cosx$ and $(sinx)^2$ If not please explain. If so please confirm. Thanks
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3answers
295 views

Showing $\left\lvert \frac{\sin(nx)}{n\sin(x)} + \frac{\cos(nx)}{n\cos(x)} \right\rvert \le\left\lvert\frac{n+1}{n}\right\rvert $

It was shown in here that $$\left\lvert \frac{\sin(nx)}{n\sin(x)}\right\rvert \le1\,\,\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$$ iff $n$ is a non-zero integer. Using the similar argument in ...
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1answer
650 views

Coefficients in Trigonometric Functions

So, previously in my math classes I have been taught that The phase shift of $$\sin(ax+c)$$ is c. But in my new class they say that $$ \sin(ax+c) = \sin(a(x+c))$$ which I believe to be wrong. But I ...
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1answer
211 views

Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose ...
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4answers
84 views

How to verify this trigonometric identity?

Please help, $9\cos^2 B − 9\sin^2 B = 18\cos^2 B − 9$ Thanks
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1answer
186 views

Minimal set of trig identities to define all the trig functions

What are a minimal set of trig identities that can uniquely define the trig functions? I know that you can define, for example, $\sin(x)$ as the unique solution to the differential equation $f''(x) = ...
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2answers
211 views

Figure out a function expression from graph (sine and cosine)

I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions So far I am sure that the gray line is ...
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2answers
320 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
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3answers
326 views

$\pi$, Dedekind cuts, trigonometric functions, area of a circle

(I should say at the outset that this question is broad, and may need splitting up. Although I ask several questions, I present them as one because they are not independent of one another, and I am ...
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2answers
84 views

what's the range of $y=\frac{\sin x+a}{\cos x+b}$ .

what's the range of $y=\frac{\sin x+a}{\cos x+b}$ . It is a question I meet somewhere, I hope to find the most simple solution.
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1answer
263 views

dividing an offset circle into triangles

First of all - I am sorry if it is the wrong forum or if this is a very trivial question. I am not a mathematician nor a trigonometry genius - and therefor I would ask a simple answer that someone ...
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1answer
106 views

Is there an easy way to see that $\left(2 \tan^2x+2\right)^3 = 8\left(\tan^2x+1\right)^3$?

I'm reading this explanation of integrals with quadratics and the author pulled this out of nowhere. Is it obvious to everyone but me that this statement is true?
3
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3answers
416 views

Trigonometry without sine and cosine

Maybe an unusual (and too easy for you) question, but my younger brother is requested to calculate the height of the Eiffel Tower: Is this possible, given that he has not learned sine and cosine ...
2
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1answer
116 views

Request for proof of the following identity.

How do we establish the following identity? \begin{align} &\int_{0}^{\frac{\pi}{2}} \frac{1}{(\sin(\theta) + \cos(\theta)) \sqrt{\sin(\theta) \cos(\theta)}} \,d{\theta} \\ \stackrel{\text{def}}{=} ...
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1answer
159 views

Trig integral 2 ways: are both equivalent?

This problem came from some other website, where someone asked for help with the integral $$\int\frac {x \, dx}{1-\cos x}$$ After adding my suggestion of integration by parts to an existing ...
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2answers
1k views

Easy ways to remember trigonometric identities

Are there any easy ways or mnemonics to memorize the trigonometric identities like for example $$ \sin(3x) = 3\sin(x) - 4\sin^3(x) $$ I find them quite difficult to come up with, I almost always need ...
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1answer
425 views

How to prove $ \sin x=…(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})…$? [duplicate]

Possible Duplicate: infinite product of sine function Here is an other one which is more or less what Euler did in one of his proofs. The function sinx where x∈R is zero exactly at ...
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0answers
262 views

flaw in law of cosines usage

I'm going to through a list of coordinates and computing the angle between every two adjacent lines. In other words, I'm computing an angle for every 3 consecutive points. Every three consecutive ...
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3answers
564 views

How do I solve $\int \sec^3 \theta d\theta$ [duplicate]

Possible Duplicate: Indefinite integral of secant cubed I guess I need to learn a new technique because those I know didn't help me here. Wolframalpha uses the reduction formula, which I ...
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1answer
131 views

find the multiplicative factor for get a specific amount of sum on sin

i am not a math guru so please sorry if this is a silly question. i'm not sure on how to latexize this question so i've done a spreadsheets with openoffice (and i'm interest also in the best way to ...
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4answers
421 views

Frequency of a trigonometric function - Where is my mistake?

I need to find the frequency of the following trigonometric function.$$y=\sin^4(x)+\cos^4(x)$$ The "answers" section says the answer is: $$F_y=\frac{\pi}{2}$$ This is what i did: Finding $\sin(x)^4$ ...
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1answer
495 views

Parameterize a straight line using polar coordinates… without angle.

I had to parameterize a straight line with starting point in $A=(-3,7)\\ $ and endpoint in $B=(4,1)$. My idea was to use the equation for the line that goes through two points. That is: $$ \frac { ...
3
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3answers
447 views

$|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$

Find all real number $n$ such that $|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$.
2
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1answer
45 views

Trigonometric bounds

Is there a nice way to show: $\sin(x) + \sin(y) + \sin(z) \geq 2$ for all $x,y,z$ such that $0 \leq x,y,z \leq \frac{\pi}{2}$ and $x + y + z = \pi$?
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2answers
176 views

Trigonometry Inequality

This is the first time I'm posting here. If you can also tell me how to format this like a pro, I'll be very grateful. 1st question: Prove the following inequality: $$0^{\circ} < a, b, c < ...
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2answers
821 views

Find the limit without l'Hôpital's rule

Find the limit $$\lim_{x\to 1}\frac{(x^2-1)\sin(3x-3)}{\cos(x^3-1)\tan^2(x^2-x)}.$$ I'm a little rusty with limits, can somebody please give me some pointers on how to solve this one? Also, ...
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2answers
622 views

What is $\cot(\pi/2)$?

Base on the unit circle, I know $ \begin{align} &\cot\left(\frac{\pi}{2}\right) \\ =&\frac{0}{1}\\ =&0 \end{align} $ But it is also $ \begin{align} &\cot\left(\frac{\pi}{2}\right) ...
4
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3answers
161 views

Proving trigonometric Identity

I would like to try and prove $$\frac{1+\sin x}{\cos x} = \frac{1+\sin x+\cos x}{1-\sin x+\cos x}$$ using $LHS=RHS$ methods, i.e. pick a side and rewrite it to make it identical to the other side. ...
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1answer
2k views

Solving a trignometric equation of form $a\sin x + b\cos x = c$

Suppose that there is a trignometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x ...
3
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1answer
386 views

Tenenbaum and Pollard, Ordinary Differential Equations, problem 1.4.29, what am I missing?

Tenenbaum and Pollard's "Ordinary Differential Equations," chapter 1, section 4, problem 29 asks for a differential equation whose solution is "a family of straight lines that are tangent to the ...
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3answers
130 views

Prove an inequality with a $\sin$ function

$$\forall{x\in(0,\frac{\pi}{2})}\ \sin(x) > \frac{2}{\pi}x $$ I suppose that solving $ \sin x = \frac{2}{\pi}x $ is the top difficulty of this exercise, but I don't know how to think out such ...
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6answers
239 views

Why isn't $\frac{d}{dx} \sin (2x) = \cos (2x)$?

$g=\sin$, $g'=\cos$ If $g(x)=\sin(x)$, then $g'(x)=\cos(x)$. Then $g(y)=\sin(y)$ and $g'(y)=\cos(y)$. Let $y=2x$. Then $g(y)=\sin(2x)$ and $g'(y)=\cos(2x)$, but if $h(x)=\sin(2x)$, then ...
5
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1answer
145 views

Let $m \in \mathbb Z, m>1$, then $\cos(2 \pi/m) \in \mathbb Q$ if and only if $m \in \{1,2,3,4,6\}$. [duplicate]

Possible Duplicate: When is $\sin(x)$ rational? Let $m \in \mathbb Z, m\geq1$, then $\cos(2 \pi/m) \in \mathbb Q$ if and only if $m \in \{1,2,3,4,6\}$. Why is this statement true? Why ...
3
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2answers
2k views

limit of inverse trigonometric function

I have the following limit that I am trying to solve but apparently I am stuck in doing l'Hôpital's rule and going nowhere so any help would be appreciated $$\lim_{x\to 0} \frac{\arcsin ...
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1answer
37 views

viewable distance from a camera stationed at a certain height (quadrotor)

I have a quadrotor with a maximum altitude of 160 feet, and a bottom mounted camera with a wide angle lens of 92 degrees. I'm wondering, at max altitude, what is the viewable land area it can see (in ...