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

learn more… | top users | synonyms (2)

12
votes
7answers
544 views

Explaining $\cos^\infty$

I noticed something odd while messing around on my calculator. $$\lim_{x\to \infty} \cos^x(c)=0.7390851332$$ Where $c$ is a real constant. My calculator is in radians and I got this number by simply ...
13
votes
2answers
2k views

When is $\sin x$ an algebraic number and when is it non-algebraic?

Show that if $x$ is rational, then $\sin x$ is algebraic number when $x$ is in degrees and $\sin x$ is non algebraic when $x$ is in radians. Details: so we have $\sin(p/q)$ is algebraic when $p/q$ ...
11
votes
1answer
3k views

A series expansion for $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
8
votes
1answer
801 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
7
votes
1answer
1k views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
6
votes
1answer
639 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
3
votes
1answer
270 views

Is there a general formula for $\sin( {p \over q} \pi)$?

Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$ And other values can be calculated through ...
13
votes
3answers
11k views

Is $\tan(\pi/2)$ undefined or infinity?

The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself. If we also observe a fraction $F=a/b$ ...
12
votes
7answers
1k views

Limit, solution in unusual way

I have a problem with solution of this limit: $$\lim_{x\to 0}{\frac{\tan{x}-x}{x^2}}$$ Of course, it's a very easy to solve, using (twice) L'Hôpital's rule, but I need to find out, how to do this ...
11
votes
2answers
3k views

Explicitly finding the sum of $\arctan(1/(n^2+n+1))$

This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$
9
votes
2answers
40k views

Limit approaching infinity of sine function

I'd like to ask a question which I have been reflecting on for some time now. What is the limit of: $f(x) = \sin(x)$ as $x$ tends to infinity? As we know, the function has a definite value for each ...
6
votes
5answers
486 views

Solve problem of trigonometry.

Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$. I have no clue how to proceed and tried to prove that the whole equation becomes $0$ when $\sin\frac{\pi}{14}$ is placed in ...
6
votes
2answers
1k views

Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} \cos{((n-...
5
votes
3answers
290 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 $-3/2$
11
votes
1answer
678 views

Proving that $ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+…+\frac{1}{\sin(133°)\sin(134°)}=\frac{1}{\sin(1°)}$

I would like to show that the following trigonometric sum $$ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+\cdots+\frac{1}{\sin(133°)\sin(134°)}$$ ...
11
votes
3answers
5k views

How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that $$1+...
9
votes
2answers
778 views

Sum of the reciprocal of sine squared

I encountered an interesting identity when doing physics homework, that is, $$ \sum_{n=1}^{N-1} \frac{1}{\sin^2 \dfrac{\pi n}{N} } = \frac{N^2-1}{3}. $$ How is this identity derived? Are there any ...
7
votes
1answer
2k views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
4
votes
2answers
169 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
4
votes
4answers
2k 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
1answer
94 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 ...
6
votes
1answer
999 views

Sine values being rational

Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
5
votes
5answers
4k views

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 + \...
4
votes
5answers
854 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.
9
votes
5answers
629 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 ...
5
votes
5answers
213 views

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$ My work: $(\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b})=\frac{1}{a+...
4
votes
2answers
3k views

Integrate $\csc^3{x} \ dx$

I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below. How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - \...
1
vote
4answers
397 views

factor $z^7-1$ into linear and quadratic factors and prove that $ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$

Factor $z^7-1$ into linear and quadratic factors and prove that $$ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$$ I have been able to prove it using the value of $\cos(\pi/7)$. Given here ...
2
votes
5answers
434 views

Prove that $\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B)$

This is my attempt: $$ \begin{align} & \phantom{={}}\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B) \\[8pt] & = (\sin(A)\cos(B)+\cos(A)\sin(B))(\sin(A)\cos(B) - \cos(A)\sin(B)) \\[8pt] & = (\...
2
votes
3answers
400 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\...
0
votes
3answers
76 views

Clarification regarding a question

In the question in the link is it compulsory that $A+B+C=\pi$ ? If sin A +sin B+sin C = cos A+cos B+cos C=0 prove that sin 2A+sin 2B+sin 2C =cos 2A+cos 2B+cos 2C
38
votes
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 2\...
38
votes
16answers
4k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
19
votes
4answers
976 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = \...
10
votes
4answers
423 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 \sin\frac{2\pi}{7}+\sin\frac{4\pi}{...
21
votes
4answers
3k 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
votes
6answers
937 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 $(...
10
votes
2answers
1k views

Evaluate $\cos 18^\circ$ without using the calculator

I only know $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ as standard angles but how can I prove that $$\cos 18^\circ=\frac{1}{4}\sqrt{10+2\sqrt{5}}$$
8
votes
3answers
2k views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
7
votes
5answers
18k views

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 ...
5
votes
2answers
130 views

If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt. [duplicate]

$$ 0<\theta<\pi/2$$ and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$ then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$ I tried to use ...
8
votes
1answer
1k views

A contradictory integral: $\int \sin x \cos x \,dx$

I've been thinking about integration lately, and I've come up with a question that I'm not sure how to address. Consider $$ \int \sin x\cos x \, dx = - \int -\sin x \cos x \, dx $$ I started with the ...
5
votes
5answers
526 views

Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{{\pi}/{2}} \log(1+\cos x)\ dx$ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
12
votes
5answers
6k views

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
votes
3answers
14k 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 ...
10
votes
3answers
433 views

Calculating :$((\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))…((\sqrt{3}+\tan(29^\circ))$

What is the easiest way to calculate : $$(\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))...((\sqrt{3}+\tan(29^\circ)) $$
7
votes
2answers
402 views

Prove that $\frac{1}{4-\sec^{2}(2\pi/7)} + \frac{1}{4-\sec^{2}(4\pi/7)} + \frac{1}{4-\sec^{2}(6\pi/7)} = 1$

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
3answers
239 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculate the definite integral $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;dx $$ given that $a>b>0$ My Attempt: If we replace $x$ by $C$, then $$ I = \int_{0}^{\pi}\...
17
votes
3answers
468 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 ...
11
votes
6answers
1k 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 ...