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 (1)

8
votes
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?
6
votes
4answers
9k 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 ...
6
votes
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 ...
4
votes
7answers
281 views

Exact value for $\cos 36°$

Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$ The problem reads as follows An ...
10
votes
2answers
4k 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
1answer
2k views

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

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
8
votes
2answers
150 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 ...
7
votes
2answers
412 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
1k 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?
11
votes
5answers
2k views

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?
7
votes
2answers
555 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} ...
4
votes
4answers
5k views

Prove $\cos 3x =4\cos^3x-3\cos x$

How would I solve the following double angle identity. $$\cos 3x =4\cos^3x-3\cos x $$ I know $\,\cos 3x = \cos(2x+x)$ So know I have $\,\cos 2x +\cos x \,$ , Which is $\,(2\cos^2x-1)\cos x$ But ...
4
votes
4answers
376 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
237 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
587 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 ...
2
votes
3answers
464 views

If $\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1$ then find the value of $\sin(c)$

If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the ...
2
votes
4answers
908 views

How to derive inverse hyperbolic trigonometric functions

$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of ...
2
votes
0answers
302 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 ...
0
votes
2answers
267 views

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

And when does $\sin^{-1}(\sin(x)) = x$
0
votes
1answer
602 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 ...
23
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 ...
9
votes
6answers
479 views

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ (Hint: set $x = \sin y$ and use implicit differentiation) So, I tried to use the hint and I got: $x = \sin y$ ...
6
votes
1answer
488 views

Rigorous proof of an infinite product.

I'll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos ...
5
votes
4answers
120 views

How to simpify $\cos x - \sin x$

How does one simplify $$\cos x - \sin x$$ I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse. Yet ...
5
votes
4answers
284 views

Trigonometric Equation $\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$

How can I solve this trigonometric equation? $$\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$$
5
votes
3answers
5k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
4
votes
3answers
1k views

How do we find specific values of sin and cos given the series definition

$\exp:x\mapsto \sum\limits_{n=0}^{+\infty}\cfrac{1}{n!}x^n$ $\cos:x\mapsto \Re\left(\exp \left(i x\right)\right)=\sum\limits_{n=0}^{+\infty}\cfrac{\left(-1\right)^n}{\left(2n\right)!}x^{2n}$ ...
4
votes
2answers
2k 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 - ...
3
votes
2answers
666 views

How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?

Given $4 \cos^2 x -2\cos x -1 = 0$. Use this to show that $\cos 36^{\circ} = (1+ \sqrt 5)/4$, $\cos 72^{\circ} = (-1+\sqrt 5)/4$ Your help is greatly appreciated! Thanks
3
votes
4answers
332 views

Given that $\;\sin^3x\sin3x = \sum^n_{m=0}C_m\cos mx\,,\; C_n \neq 0\;$ is an identity . Find the value of n.

Problem : Given that $\sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx, C_n \neq 0 $ is an identity. Find the value of n. I tried : $\sin3x = 3\sin x - 4\sin^3 x$ but unable to reach to any point.... ...
3
votes
4answers
772 views

Sequence of solutions to $x\sin x=1$

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Consider a sequence $x_n, n\ge1$ formed by positive solutions to ...
2
votes
3answers
103 views

Convergence test of the series $\sum\sin100n$ [duplicate]

I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
1
vote
2answers
50 views

How to to a better approach for this :?

If, $$x\cos A+y\sin A=k=x\cos B+y\sin B$$ Then find $(\cos A)(\cos B)$, $(\sin A)(\sin B)$ and $\cos A+\cos B$ and express them in terms of $x,y,k$ I found a solution but it included a really ...
1
vote
4answers
168 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 ...
1
vote
2answers
87 views

Relationship among $A,B,C,D$ for $\cos A\cos B=\cos C\cos D$

While solving this Question, I could derive the following: As $\displaystyle 2\cos A\cos B=\cos(A-B)+\cos(A+B)$ substituting $A+B=90^\circ\iff B=90^\circ-A$ we get $\displaystyle 2\cos ...
8
votes
2answers
259 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
6
votes
3answers
441 views

Prove trigonometry identity for $\cos A+\cos B+\cos C$

I humbly ask for help in the following problem. If \begin{equation} A+B+C=180 \end{equation} Then prove \begin{equation} \cos A+\cos B+\cos C=1+4\sin(A/2)\sin(B/2)\sin(C/2) \end{equation} How would ...
4
votes
3answers
254 views

Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?

Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$? Can someone please help me with question, im very lost.
4
votes
4answers
2k views

Trying to derive an inverse trigonometric function

I'd like to know how to derive these functions (I know the answers, I want to know how to get there) \begin{align*} f(x) &= \arcsin\left(\frac{x}{3}\right)\\ f(x) &= \arccos(2x+1)\\ f(x) ...
3
votes
2answers
895 views

Finding the widest angle to shoot a soccer ball from the sideline using optimization

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
3
votes
2answers
319 views

Prove that $x^2<\sin x \tan x$ as $x \to 0$ [duplicate]

$$x^2<\sin x \tan x \quad as \; x \to 0$$ I made the substitution $x \to \arctan x$ . $\arctan^2 x<x\sin (\arctan x)$ $\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$ There are two ...
2
votes
4answers
180 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
2
votes
3answers
141 views

Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ [closed]

Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$? The correct answer should be 4.
2
votes
2answers
81 views

Find the product

Task is to find $$\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$$ I tried to make it with double-angle formula: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}) = \frac{\prod_{k=0}^\infty ...
2
votes
1answer
360 views

Tide and Trigonometric functions

I have a tide guide that gives me four readings for the day - 2 high tides and two low tides. This means it completes two full revolutions within a day. What I'm having trouble with is taking the four ...
0
votes
1answer
38 views

Finding equation using hyperbolic transcendental functions.

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
0
votes
1answer
200 views

Trigonometric AP relation on sides of a triangle

The sides of a triangle are in AP (Arithmetic Progression) and the greatest angle exceeds the least angle by $90$ degrees prove that the sides are proportional to $7^{\frac{1}{2}}+1$ , ...
0
votes
1answer
92 views

Help needed with trigonometric identity

Prove that $$\left(\sqrt{3} -4\sin\left(\frac{2\pi}{15}\right)\right)\cos\left(\frac{\pi}{30} \right) =\sin\left(\frac{\pi}{30} \right).$$
0
votes
1answer
92 views

Engineering Mathematics

I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. I cannot envisage the compound angle ...
-1
votes
1answer
59 views

Unordered pairs solution

Please help me with this question.$$$$ How many unordered triplets $(x,y,z)$ , subject to constraints, $(x^4-2x^3)_{cyclic}\leq0$ , satisfy the system of equations: ...