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

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12
votes
4answers
300 views

Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$

Or put differently, does $$\lim_{n \to \infty}\big(\max \{\sin 1, \sin 2, \ldots ,\sin n\}\big) = 1?$$ My intuition says yes, but how can one prove this?
10
votes
2answers
327 views

A matrix w/integer eigenvalues and trigonometric identity

Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated: Let $n$ be a natural number. (a) Consider the following Toeplitz/circulant symmetric matrix: ...
8
votes
3answers
316 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 ...
7
votes
4answers
349 views

Nth derivative of $\tan^m x$

$m$ is positive integer, $n$ is non-negative integer. $$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$ $P_n(x)=f_n(\arctan(x))$ I would like to find the polynomials that are defined as above ...
6
votes
2answers
791 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
6
votes
4answers
976 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 ...
4
votes
2answers
133 views

In triangle, $\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$

To prove $$\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$$ My approach : $$ \begin{align} \text{L.H.S.} & = ...
2
votes
1answer
132 views

A question by Ramanujan about a relational expression of a triangle

I found the following question in a book without any proof: Question : Suppose that each length of three edges of a triangle $ABC$ are $BC=a, CA=b, AB=c$ respectively. If $$\frac1a=\frac1b+\frac1c, ...
18
votes
2answers
437 views

integrate square of $\arctan x$. Tricky

$$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$ I ran across an integral I am having a time solving. The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
10
votes
2answers
10k views

How does a calculator calculate the sine, cosine ,tangent using just a number?

Sine Θ = oposite/hypotenuse Cosine Θ = adjacent/hypotenuse Tangent Θ = oposite/adjacent So in order to calculate the Sine or the cosine or the tangent I need to ...
9
votes
3answers
239 views

How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$

Is there an easy way to prove the identity? $$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3} \right ...
8
votes
10answers
3k views

How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?.

How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ without using a calculator. Related question: how do we prove that $\cos(\pi/5)\cos(2\pi/5) = 0.25$, also without using a calculator
8
votes
2answers
222 views

Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$

My Question is: Examine the continuity of $$f(x) = \lim_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \qquad \text{for} \ x \in \mathbb{R}$$ How can I do this? Honestly, speaking I have $\text{no ...
7
votes
2answers
341 views

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number. I heard that this was proved two hundred years ...
5
votes
3answers
149 views

minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$

How to prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$
4
votes
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 $100\times 100$ ...
3
votes
1answer
51 views

Equivalence of the two cosine definitions

There are at least two ways to define the cosine function: You can define it with a right triangle in the unit circle and extend the definition to $\mathbb{R}$. (classic definition) The other ...
3
votes
4answers
2k views

Memorizing the unit circle?

I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the ...
1
vote
2answers
222 views

Finding an unknown angle

Geometry: Auxiliary Lines As shown in the figure:
21
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
11
votes
4answers
785 views

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$ I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
11
votes
5answers
692 views

Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$

I am trying to study for a test and the teacher suggest we memorize $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, $\cos (A-B)$, and then be able to derive the rest out of those. I have no idea how to get ...
10
votes
6answers
671 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
10
votes
2answers
2k views

What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
9
votes
4answers
1k views

Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...
7
votes
1answer
209 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
6
votes
3answers
854 views

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$

I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me: Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers ...
5
votes
1answer
1k views

Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]

For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
4
votes
4answers
994 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, ...
3
votes
3answers
117 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
3
votes
4answers
600 views

$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}$ without using L'Hopital

$$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$$ I tried using $\lim_{x\to0} \frac{\sin x}{x}=1$. But it doesn't work :/
3
votes
4answers
229 views

Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$

I am solving this problem. If $$\sum\limits_{i=1}^{\infty} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t$$ then find the value of $\tan{t}$. My solution is like the following: I can rewrite: ...
2
votes
1answer
117 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 ...
1
vote
4answers
153 views

Difficult trigonometric equation

Please, can you suggest something for solving this equation: I have to find the solutions included in interval $\left[3\pi/2, 2\pi\right]$: $$\sqrt{3\cos^2 x - \sin 2x} = - \sin x$$ This is what I ...
0
votes
2answers
207 views

Why does $\sin^{-1}(\sin(\pi))$ not equal $\pi$

And when does $\sin^{-1}(\sin(x)) = x$
13
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
10
votes
3answers
207 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
10
votes
2answers
467 views

Evaluating $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

I have to evaluate: $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx $$ I can't get the right answer! So please help me out!
9
votes
1answer
313 views

Calculating $\sin(10^\circ)$ with a geometric method

Excuse me if this is a simple question: What is a simple geometric method for calculating $\sin(10^\circ)$ using only the sines of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$? Generally, is ...
9
votes
3answers
7k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
7
votes
1answer
247 views

Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$

I would like to show that: $$ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$$ We have: $$ \sum_{n=1}^N \arctan \left( \frac{2}{n^2} \right) =\sum_{n=1}^N \arctan ...
7
votes
4answers
449 views

Intuition for graphing Sine/Cosine

So I'm working my way through some basic trig (Khan Academy) - I'm trying to get a better intuition for what graphs of sine and cosine represent. I've seen the nice unit circle animations that do ...
7
votes
2answers
242 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 ...
7
votes
5answers
2k 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?
6
votes
2answers
71 views

Another trigonometric proof…?

...sigh..another problem how shall I prove the following? $$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$ so what now? the following's what I've done: $$\cot A - \cot^2 A ...
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...
5
votes
2answers
484 views

Method to find $\sin (2\pi/7)$

I just thought a way to find $\sin\frac{2π}{7}$. Considering the equation $x^7=1$ $⇒(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$ $⇒(x-1)[(x+\frac1 x)^3+(x+\frac1 x)^2-2(x+\frac1 x)-1]=0$ We can then get the ...
5
votes
4answers
592 views

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
5
votes
2answers
269 views

Prove $\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$

How can I prove the fact $$\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$$ When asked somebody told me to use the ideas of ...
5
votes
2answers
516 views

Proof about $z\cot z=1-2\sum_{k\ge1}z^2/(k^2\pi^2-z^2)$

In Concrete Mathematics, it is said that $$z\cot z=1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}\tag1$$ and proved in EXERCISE 6.73 $$z\cot z=\frac z{2^n}\cot\frac z{2^n}-\frac z{2^n}\tan\frac ...