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

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Solving $ \sin x + \sqrt 3 \cos x = 1 $ - is my solution correct?

I have an equation that I'm trying to solve: $$ \sin x + \sqrt 3 \cos x = 1 $$ After pondering for a while and trying different things out, this chain of steps is what I ended up with: $$ \sin x + ...
5
votes
2answers
846 views

Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$

how to prove $\sup \{ \sin n \mid n\in \mathbb N \} =1$
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5answers
745 views

How to prove a trigonometric identity

Show that $$ \tan(A)=\frac{\sin2A}{1+\cos 2A} $$ I've tried a few methods, and it stumped my teacher.
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3answers
147 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
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4answers
889 views

Is there a more efficient method of trig mastery than rote memorization?

I would like to get alot better at trig than I am. What is the best/most efficient method? Thanks much in advance Joe
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5answers
496 views

How to see $\sin x + \cos x$

$$\sin x + \cos x = \sqrt{2} \sin(x + \pi/4)$$ Is there an easy way to visualize this identity or to convert the left-hand side to the right-hand side? In general, can $p \sin x + q \cos x$ for ...
6
votes
3answers
279 views

Proof that $\lim_{n\to\infty}{\sin{100n}}$ does not exist

How to prove that $$\lim_{n\to\infty}{\sin{100n}}$$ doesn't exist? Some possible approaches: It would be enough to find two subsequences $n_{k}$ that converge to two different numbers. But ...
6
votes
2answers
749 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
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1answer
628 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
4
votes
1answer
149 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
4
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6answers
796 views

How to find the exact value of $ \cos(36^\circ) $?

The problem reads as follows: Noting that $t=\frac{\pi}{5}$ satisfies $3t=\pi-2t$, find the exact value of $$\cos(36^\circ)$$ it says that you may find useful the following identities: ...
4
votes
4answers
650 views

What is the most elegant and simple proof for the law of cosines?

Given 2 sides, and an angle between those two sides, what is the simplest proof you can come up with to find the measure of the 3rd side?
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3answers
2k views

How do I measure distance on a globe?

I have a $3$-D sphere of radius $R$, centered at the origin. $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are two points on the sphere. The Euclidean distance is easy to calculate, but what if I were to ...
3
votes
4answers
370 views

Find the value of $\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right)$

What is the value of $$\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right) \qquad\qquad ? $$ I tried to write ...
5
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2answers
331 views

Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$

Assume we have: $ \int{ \cos{x} + \sin{x}\cos{x} dx } $ 2 ways to do it: Use $\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $ Then $ \int{ \cos{x} + \frac{\sin{2x}}{2} dx } $ $ = \sin{x} - \frac{ cos{2x} ...
3
votes
2answers
113 views

Why does $A\sin{k(x+c)}=a\sin{kx}+b\cos{kx}$ imply that $A=\sqrt{a^2+b^2}$ and $\tan{c}=-b/a$?

I don't understand this. These identities are given in the online notes for MIT's 18.01 calculus class. It's related to taking the sum of two trig functions and transforming them into a single trig ...
3
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2answers
194 views

Proving trigonometric equation $\cos(36^\circ) - \cos(72^\circ) = 1/2$ [duplicate]

Please help me to prove this trigonometric equation. $\cos \left( 36^\circ \right)-\cos \left( 72^\circ \right) = \frac{1}{2}$ Thank you.
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3answers
2k views

How can I find the derivative of $y = \sin(\arctan x) + \tan(\arcsin x)$?

My question is, how can I solve the following derivative question? $$y = \sin(\arctan x) + \tan(\arcsin x)$$ Thanks in advance
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2answers
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Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
33
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2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
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votes
5answers
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Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this ...
38
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1answer
847 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
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8answers
6k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
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4answers
2k views

Is there a limit of cos (n!)?

I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler ...
14
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2answers
858 views

Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$

How can one show that the number $2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$ is a root of the equation $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
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4answers
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Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
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3answers
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Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
19
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8answers
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Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
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votes
4answers
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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 ...
14
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6answers
710 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 ...
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3answers
267 views

A trigonometric equation

Some years ago, I came across the following question: Find the value of $\tan5°\tan55°\tan65°\tan75°$. The value is 1 and I generalized the question as follows. $\tan a°\tan b°\tan c°\tan d° ...
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1answer
3k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
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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 ...
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9answers
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Good book recommendations on trigonometry

I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line: Now that you know how the circular functions are defined, you might wonder how ...
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votes
4answers
273 views

Trigo Problem : Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$

Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ My approach : I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $ $\Rightarrow ...
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7answers
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Exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$

What are the exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$ and how do I work it out? I know that $\cos(2\pi/7)$ and $\sin(2\pi/7)$ are the real and imaginary parts of $e^{2\pi i/7}$ but I am not ...
8
votes
2answers
137 views

Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$

The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...
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1answer
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A series expansion of $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
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2answers
401 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 ...
6
votes
1answer
992 views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
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4answers
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Determine third point of triangle when two points and all sides are known?

Determine third point of triangle (on a 2D plane) when two points and all sides are known? A = (0,0) B = (5,0) C = (?, ?) AB = 5 BC = 4 AC = 3 Can someone ...
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1answer
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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 ...
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5answers
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Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
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3answers
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Why is it that when proving trig identities, one must work both sides independently?

Suppose that you have to prove the trig identity: $$\frac{\sin\theta - \sin^3\theta}{\cos^2\theta}=\sin\theta$$ I have always been told that I should manipulate the left and right sides of the ...
10
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2answers
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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 ...
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2answers
473 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
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votes
4answers
1k 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 :/
4
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4answers
356 views

A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere. $$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$ Could someone please provide a valid ...
3
votes
2answers
230 views

Find this limit without using L'Hospital's rule

I have to find this limit without using l'Hôspital's rule: $$\lim_{x\to 0} \frac{\alpha \sin \beta x - \beta \sin \alpha x}{x^2 \sin \alpha x}$$ Using L'Hôspital's rule gives: ...
3
votes
2answers
581 views

Proving: $\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A … \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $

$$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$ I am very much inquisitive to see how this trigonometrical identity can be ...