Questions tagged [trigonometry]
Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.
29,694
questions
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Proof for $\sin(x) > x - \frac{x^3}{3!}$
They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried:
$\sin(...
- 747
22
votes
1
answer
572
views
How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$? [closed]
How can I prove the following identity?
$$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
- 13.1k
22
votes
2
answers
2k
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Asymptotic behaviour of sum
I would like to evaluate the number $c$ given by
$$
c = \lim_{m\to\infty} \frac{1}{\log m}\sum_{n=1}^m \frac{1}{n^2 \sin^2(\pi n \tau)}
$$
where $\tau = (1+\sqrt{5})/2$.
My attempt: my guess was ...
22
votes
1
answer
673
views
Some trigo identities
I aacidently found the following:
$$\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}-\sin\frac{6\pi}{7}=\frac{\sqrt{7}}{2}$$
$$\sin\frac{2\pi}{11}-\sin\frac{4\pi}{11}+\sin\frac{6\pi}{11}+\sin\frac{8\pi}{11}+\sin\...
- 2,461
22
votes
3
answers
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Explaining the graph of $\sin(x^2) + \sin(y^2) = 1$
I had to plot the graph of the implicitly defined function $\sin^2 x + \sin^2 y = 1$ in an exam. This is not particularly difficult, but it got me wondering what the graph would look like when the ...
- 677
22
votes
2
answers
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an integer sum of products of tangents
This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that ...
- 238k
22
votes
3
answers
3k
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Existence of continuous angle function $\theta:S^1\to\mathbb{R}$
Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function
$$\theta:U\to\mathbb{R}$$
such that
$$e^{i\theta(z)}=z$$
for all $z\...
- 221
22
votes
1
answer
460
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Generalizing two infinite products for $\operatorname{sinc}(x)$ and their 'dual' infinite product
$\newcommand{\sinc}{\operatorname{sinc}}$
Throughout, let $m,k$ be positive integers, $x>0$ a real number, and denote $\sinc(z)=\sin(z)/z$ with $\sinc(0)=1$.
A famous result of Euler gives $\sinc(x)...
- 8,467
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votes
0
answers
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How to find the approximate basic period or GCD of a list of numbers?
I want to tell the number which act as the best approximate basic period (or wavelenght as pointed out by Eric) of a list of real numbers: e.g for {14, 21, 35} we should obtain 7 as the basic period, ...
- 366
21
votes
14
answers
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Find $4\cos\theta-3\sin\theta$, given that $4\sin \theta +3\cos \theta = 5$ [closed]
Another problem that I already wasted hours on.
Given
$$4\sinθ +3\cosθ = 5$$
Find
$$4\cosθ -3\sinθ$$
Help me guys (PS:I'm not that good in maths)
- 1,420
21
votes
8
answers
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views
what is sine of a real number
I never understand what the trigonometric function sine is..
We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. Till ...
user311526
21
votes
2
answers
2k
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Evaluating $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x$
I have to evaluate:
$$\int_{0}^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x. $$
I can't get the right answer! So please help me out!
- 3,165
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7
answers
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Proving $\sum_{k=0}^{n}\cos kx=\frac{1}{2}+\frac{\sin(n+\frac12)x}{2\sin\frac x2}$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
- 23.2k
21
votes
7
answers
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views
Relationship between $\tanh x$ and $\arctan x$
The functions $\tanh x$ and $\arctan x$ have a similar graph. Is there a formula to transform $\tanh x$ to $\arctan x$?
- 609
21
votes
7
answers
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Solving trigonometric equations of the form $a\sin x + b\cos x = c$
Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos ...
- 233
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2
answers
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Is this an incorrect proof of $\cot (x)+\tan(x)=\csc(x)\sec(x)$?
If you input the trig identity:
$$\cot (x)+\tan(x)=\csc(x)\sec(x)$$
Into WolframAlpha, it gives the following proof:
Expand into basic trigonometric parts:
$$\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\...
- 4,934
21
votes
1
answer
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Finding the angle between two points
First of all, I am doing some mathematical background information for a software I am creating.
What I want to achieve is the point on an object rotating towards where the mouse is. Like in tank ...
- 367
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votes
3
answers
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Closed form solution to $\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$
I need to compute this integral:
$$\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$$
I tried integration by parts, and also introducing a parameter $\arctan(a\,x)$ and differentiation wrt it, but these approaches ...
- 1,475
21
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5
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Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$
$$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$
My approach :
Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then
$$\int \frac{x^2\sec^2x}{...
- 9,906
21
votes
1
answer
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$\arcsin$ written as $\sin^{-1}(x)$
I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not ...
- 9,512
21
votes
8
answers
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views
Mental estimate for tangent of an angle (from $0$ to $90$ degrees)
Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable.
I can ...
- 837
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votes
6
answers
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Does $\sum_{n=1}^\infty \frac{\cos{(\sqrt{n})}}{n}$ converge?
The series is:
$$\sum_{n=1}^\infty \frac{\cos(\sqrt{n})}{n}$$
Considering it isn't always positive, I replace $\frac{\cos{\sqrt{n}}}{n}$ with its absolute value and I find that:
$$\vert \frac{\cos{\...
- 259
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votes
3
answers
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Is there a simple-ish function for modeling seasonal changes to day/night duration and height of the sun?
I'm a hobbyist programmer, and not much of a mathematician. I'm trying to model something like the seasonal change in day length. There are two other questions here that are very similar to mine, and ...
- 265
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votes
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Solving Triangles (finding missing sides/angles given 3 sides/angles)
What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
- 36.6k
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Prove a trigonometric identity: $\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1$ when $A+B+C=\pi$
There is a trigonometric identity:
$$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C\equiv 1\text{ when }A+B+C=\pi$$
It is easy to prove it in an algebraic way, just like that:
Then, I want to find a ...
- 4,033
21
votes
4
answers
345
views
Solve for $x$ a trigonometric equation
I want to solve for $x$
$$ {{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x $$
but I don't know how to start. Replacing $\sin x$ or $\cos x$ by $y$ led me nowhere ...
- 211
21
votes
2
answers
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Sequences where $\sum\limits_{n=k}^{\infty}{a_n}=\sum\limits_{n=k}^{\infty}{a_n^2}$
I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form.
I then wondered ...
- 2,796
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votes
4
answers
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views
How is $\sin 90° = 1$ possible?
How can two angles of a triangle be equal to $90°$? If two angles were $90°$, this would mean that the two sides would be parallel and the angle of the third side would be equal to 0. Thus, there ...
21
votes
2
answers
516
views
Polar plots of $\sin(kx)$
The plots of $\sin(kx)$ over the real line are somehow boring and look essentially all the same:
For larger $k$ you cannot easily tell which $k$ it is (not only due to Moiré effects):
But when ...
- 18.2k
21
votes
2
answers
678
views
To whom do we owe this construction of angles and trigonometry?
I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
- 6,766
21
votes
1
answer
685
views
Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?
When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way?
Also, is there ...
- 2,455
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votes
3
answers
772
views
Evaluating $\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}} \,dx$
How to integrate?
$$\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}}\, dx$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
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0
answers
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Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$ [closed]
Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$
I tried using by parts and complex numbers along with series expansion but I was unable to find the answer.
Please Help!
- 5,589
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votes
9
answers
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Solve $\cos^2x-\sin^2x= 1$
I am trying to solve for $x$ in the following equation:
$$\cos^2x-\sin^2x= 1$$
Given that $\cos^2x+\sin^2x= 1$, is this something I could use to solve it?
- 2,297
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votes
7
answers
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Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
How would I verify the following double angle identity.
$$
\sin(A+B)\sin(A-B)=\sin^2A-\sin^2B
$$
So far I have done this.
$$
(\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B)
$$But I am not sure ...
- 221
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votes
3
answers
4k
views
A definite integral with trigonometric functions: $\int_{0}^{\pi/2} x^{2} \sqrt{\tan x} \sin(2x) \, \mathrm{d}x$
How could we get a closed form for the following integral:
$$\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x
\right) \, \mathrm{d}x} \tag{1}$$
While the antiderivative of $\sqrt{\...
- 2,241
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votes
3
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963
views
$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$
How can I find the following product using elementary trigonometry?
$$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$
I have tried using a substitution, but nothing has ...
- 349
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votes
5
answers
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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 ...
- 1,493
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votes
5
answers
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How is the angle between 2 vectors in more than 3 dimensions defined?
I would like to know how the angle between two n-vectors is defined. I mean whether it is unique and how we may compute it (is the inner product a valid method in the n-dimensional space?). I have ...
- 201
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votes
5
answers
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Showing $\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}$
How can one show :
$$\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}?$$
- 4,896
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votes
3
answers
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Why is this trigonometric identity true?
Suppose $$\frac{{{{\sin }^4}(\alpha )}}{a} + \frac{{{{\cos }^4}(\alpha )}}{b} = \frac{1}{{a + b}}$$ for some $a,b\ne 0$.
Why does $$\frac{{{{\sin }^8}(\alpha )}}{{{a^3}}} + \frac{{{{\cos }^8}(\alpha ...
- 953
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votes
9
answers
2k
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Ways to prove $ \int_0^\pi \frac{\sin^2 nx}{\sin^2 x} dx= n\pi$
In how many ways can we prove the following theorem?
$$I(n):= \int_0^\pi \frac{\sin^2 nx}{\sin^2 x} dx= n\pi$$
where $n$ is a nonnegative integer.
The proof I found is by considering $I(n+1)-I(n)$, ...
- 5,856
20
votes
3
answers
937
views
Finding $ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$ [closed]
I would appreciate if somebody could help me with the following problem. How can we find the product
$$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$
- 5,494
20
votes
4
answers
832
views
Is there no formula for $\cos(x^2)$?
I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance $$\...
- 23.8k
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votes
4
answers
8k
views
About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$
Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.
...
- 16k
20
votes
2
answers
1k
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 $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
- 14.1k
20
votes
4
answers
3k
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Machin's formula and cousins
There exists a well-known formula by John Machin:
$$\frac{\pi}{4} = 4 \arctan \left(\frac{1}{5}\right) - \arctan \left(\frac{1}{239}\right).$$ Actually, it belongs to the family of Machin-like ...
- 549
20
votes
2
answers
5k
views
How does the Herglotz trick work?
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions ...
- 18.5k
20
votes
2
answers
2k
views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/...
user58512
20
votes
3
answers
632
views
How do I prove that $\left( 1-\frac{a}{b} \right)\left( 1+\frac{c}{d} \right)=4$?
The challenge is to prove $$\left(1-\dfrac{a}{b}\right)\left(1+\dfrac{c}{d}\right)=4.$$
Apart from the Pythagorean theorem, the tangent and secant theorem, and the cosine theorem, I could not invent ...
- 2,344