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

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7
votes
5answers
54 views

Proving that $\cos(\frac{\arctan(\frac{11}{2})}{3}) = \frac{2}{\sqrt{5}}$

I am trying to solve the cubic equation $x^3-15x-4=0$ using Cardano's formula. I already know that the solutions are $x=4$, $x= \sqrt{3}-2$ and $x= -\sqrt{3}-2$ and that using the formula in this ...
0
votes
1answer
10 views

A good way to approximate $\cos(x)$ for larger angles

I usually don't see any expressions for approximate calculation of trigonometric functions aside from their Taylor series. But Taylor series work better for small angles, so in general they are not ...
0
votes
1answer
18 views

What trigonometric identity makes the method of triangulation work?

I've read the article on Wikipedia, but I don't get how to construct the relationships between sides and angles to reach a solution for the distance between two points. All the other sites I read ...
2
votes
1answer
21 views

Trigonometric integrals and limits

Show $$\lim_{N\to\infty}g_N(\theta_N)=2\int^\pi_0\frac{\sin x}{x}dx-\pi,$$ where $$g_N(\theta_N)=\int_0^{\theta_N}\frac{\sin[(N+1/2)x]}{\sin(x/2)}dx-\pi,$$ $$\theta_N=\frac{\pi}{N+1/2},$$ and ...
1
vote
0answers
32 views

Ratios of the major axes of ellipses inscribed in a triangle

I recently came across a question that went something like this: In a triangle with vertices (0,0), (6,0), (2,4) an ellipse is inscribed such that it has the largest area. Now it is dilated, that is ...
2
votes
2answers
32 views

The correct half angle formula?

It is well known that $$\cos(\frac x2)=\sqrt{\frac{1+\cos(x)}2}$$ And, we also know that $\cos(\frac x2)$ may be negative for some $x$ values. So that implies that: $$\cos(\frac ...
0
votes
0answers
29 views

Does $\lim\limits_{x\to0}x^2\csc\frac{1}{\sqrt[3]x}$ exist?

The original problem is computing the limit $$L=\lim_{x\to1}\frac{(x-1)^2}{\sin\frac{1}{\sqrt[3]{x-1}}}$$ for which I replaced $x-1$ with $x$. Is there something wrong with invoking the limit ...
3
votes
1answer
61 views

calculate $\frac{1}{\sin(x)} +\frac{1}{\cos(x)}$ if $\sin(x)+\cos(x)=\frac{7}{5}$

If $$\sin(x)+\cos(x) = \frac{7}{5}$$ Then what's the value of $$\frac{1}{\sin(x)} +\frac{1}{\cos(x)}\ \ \text{?}$$ Meaning the value of $\sin(x)\:$, $\cos(x)$ (the denominator) without using the ...
3
votes
6answers
70 views

How to proceed from $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$

To prove: $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$ My attempt at the solution: ...
0
votes
1answer
17 views

Drawing half a circle betweeen two arbitrary 2D points

So, I have two arbitrary points in a vector space and I'm trying to draw 180 degrees of a circle between them. The radius of the circle would be half of the distance between the two points and the ...
-3
votes
1answer
33 views

solving polar simultaneous equations [on hold]

I need to solve the below polar equations. Question: Find all the points of intersections between the two polar curves s(θ)=(12θ,θ), r(θ)=(θ,12θ) Thanks
1
vote
2answers
18 views

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$ So $x\cot x = x\left( \frac{e^{ix}+e^{-ix}}{2}\cdot \frac{2i}{e^{ix}-e^{-ix}} \right) = \frac{ix(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}=\dots$ ...
0
votes
0answers
33 views

A trigonometric equation - Looking for closed form solutions.

I was wondering if the following equation has a solution \begin{equation} \frac{\sin\big[(N+1)\phi\big]}{\sin\big[N\phi\big]}=1+\frac{\alpha}{\cos\phi+\beta} \end{equation} where $\alpha$ and ...
2
votes
1answer
42 views

How can I find an output of this function's inverse without graphing?

How can I find $f^{-1}(5)$ where $$f(x)=\frac{27}{\pi}x + \sin x$$ algebraically? Thank you!!
1
vote
1answer
19 views

Calculate most parallel vector

Let's suppose I have a vector $\vec{a}$ and arbitrary many vectors (in my example two: $\vec{b}$ and $\vec{c}$), which all have a common starting point P. Now I want to find out which of these vectors ...
-1
votes
1answer
44 views

General and principal solutions of Sinx + Sin3x + Sin5x = 0?

A solution that i found on net but cant figure out why we need to do Sinx + Sin5x Why does we take the sum of Sin x and Sin 5x ? Why cant we take the sum of Sin 3x and Sin 5x or Sin x and Sin 3x? ...
3
votes
3answers
56 views

Solution of the equation $\cot \theta = 2\cot 2\theta$

I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a ...
1
vote
1answer
31 views

If $A+B+C=π$, verify the given

If $A+B+C=π$, prove that $$\cos A \sin B \sin C + \cos B \sin C \sin A + \cos C \sin A \sin B=1+\cos A \cos B\cos C$$ ATTEMPT: Here, $$A+B+C=π$$ Now, \begin{align*} \text{L.H.S} &= \cos A \sin B ...
0
votes
4answers
61 views

Hint: $\sin^2(6x)-\sin^2(4x) = \sin(2x)\sin(10x)$ [duplicate]

I need a hint to solve prove $\sin^2(6x)-\sin^2(4x) = \sin(2x)\sin(10x)$ I tried several solutions, including taking $(\sin(6x)+\sin(4x))(\sin(6x)-\sin(4x))$ but every time I ended up with a ...
-4
votes
1answer
24 views

Find both the diagonals with area and side given [on hold]

Side of rhombus $= 65$ cm Area of rhombus $= 1024$ cm$^2$ Find the diagonals of the rhombus.
3
votes
0answers
55 views

Want to know what's wrong?

I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ...
0
votes
0answers
33 views

An inequality with the sinus in a triangle.

I have solved this problem in a way, rather "inspired". I would like to have a solution found an easier way but I was unable so far. Let $A,B,C$ the angles of a triangle $\triangle {ABC}$; prove ...
4
votes
1answer
90 views

Is $\arctan2$ irrational? [duplicate]

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
1
vote
1answer
37 views

the maxima of given function

What's the maxima of $$2^{\sin(x)}+2^{\cos(x)}$$ I found max by taking logs and then differentiating and equating to $0$ at $x=45°$ so the answer is $2^{\frac{\sqrt{2}+1}{\sqrt{2}}}$ am I right or I ...
2
votes
2answers
39 views

Find the value of $ sin(2\theta)$ when $cot(\theta) + tan(\theta) = 2.5 $

I have an homework question that goes like: $cot(\theta) + tan(\theta) = 2.5 $ is valid on some angles $\theta$ at section $0 < \theta < \pi/2$. Find the value of $sin(2\theta)$. (There is no ...
0
votes
1answer
11 views

Having trouble drawing an arc using atan2 and specified range of arc.

I'm writing code. My arc invariant is: $$ \theta_0, \theta_1 \in [-2\pi, 2\pi], \\ \theta_1 - \theta_0 \leq 2\pi $$ where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle. ...
3
votes
1answer
32 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
0
votes
0answers
9 views

Find largest value of the constants that makes the equality true

$$sin(h^2) = a + O(h^b)$$ Find the largest value of a and b such that the equality holds. I tried to use the truncated Taylor series expansion of $sin(h^2)$, but the derivatives of the function are to ...
1
vote
0answers
32 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
0
votes
0answers
28 views

plotting circles of variable size on circumference of a circle

I am writing a layout algorithm for graph (i.e. vertices and edges) data and want to implement a slight twist on a hive plot. I would like to be able to draw each node as a circle (though different ...
0
votes
2answers
29 views

trigo substitution and identites?

When I use trigo substitution to solve an integral I get an expression like that: $$\frac{1}{4}\tan\left(\arcsin\left(\frac{x-2}{2}\right)\right)+C$$ How can I simplify it?
0
votes
1answer
55 views

Challenge in trignometry and integration [on hold]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
0
votes
2answers
40 views

The obtuse angle $B$ is such that $\tan B = -\frac{5}{12}$. Find the exact value of $\sin B$.

The obtuse angle $B$ is such that $\tan B = -\frac{5}{12}$. Find the exact value of $\sin B$. Is this possible to do without a calculator? If so, how?
0
votes
5answers
79 views

Solving Trigonometric Equation.

Solve for $\theta$ $[0°<\theta<180°]$ $$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$ My solution is here: $$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$ After using the ...
0
votes
3answers
52 views

Trigonometric Idntities under the condition $A+B+C=π$

If $$A+B+C=\pi$$, prove that $$\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}=4\cos\frac{\pi-A}{4}.\cos\frac{\pi-B}{4}.\cos\frac{\pi-C}{4}$$. My solution: Here $$A+B+C=\pi$$ $$A+B=\pi-C$$ Taking ...
1
vote
1answer
19 views

Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer. Show that $$T_2 (x) = 2x^2 – 1.$$ My attempt: $T_2 (x) = \cos (2 \arccos x)$ ...
3
votes
1answer
100 views

Do any mathematican still reserach about trigonometry?

Do any mathematican still reserach about applied trigonometry? If so, what are the subject area called in the PhD level except fourier analysis? In many area, you could see a lot of trig and ...
1
vote
1answer
47 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A\cos(\omega t + \phi)$ $A > 0$ $\phi\in(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes $0.123$ ...
-1
votes
4answers
82 views

If $A+B+C=π$, prove that

If $A+B+C=π$, prove that $$\cos^2A+\cos^2B-\cos^2C=-2\cos A\cdot\cos B\cdot\cos C.$$ ATTEMPT: Given $$A+B+C=π,$$ $$A+B=π-C$$ Taking "cos" on both sides $$\cos(A+B)=-\cos C.$$ Now, ...
2
votes
1answer
42 views

How would one evaluate $\sin(72\pi/11)$?

How would one evaluate $\sin(\frac {72\pi} {11})$?. The prime number in the bottom is getting me stuck. I couldn't see how to use it using the sum of two angles trig identity.
0
votes
2answers
26 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
2
votes
0answers
40 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
0
votes
3answers
49 views

If $\frac{m+1}{m-1}=\frac{cos(\alpha-\beta)}{sin(\alpha+\beta)}$„ then

If $$\frac{m+1}{m-1}=\frac{\cos(\alpha-\beta)}{\sin(\alpha+\beta)}$$, prove that : $$m=\tan(π/4 +\alpha).\tan(π/4 +\beta)$$. My attempts/ Here .. ...
1
vote
3answers
27 views

Prove that $m\tan (\theta-30°)=n\tan (\theta+120°)$

If $m\tan (\theta-30°)=n\tan (\theta+120°)$ then prove that : $$\cos 2\theta=\frac{m+n}{2(m-n)}$$ My attempt\ Here, $$m\tan (\theta-30°)=n\tan (\theta+120)$$ $$\frac{\tan (\theta-30°)}{\tan ...
2
votes
4answers
62 views

Proving $\tan A=\frac{1-\cos B}{\sin B} \;\implies\; \tan 2A=\tan B$

If $\tan A=\dfrac{1-\cos B}{\sin B}$, prove that $\tan 2A=\tan B$. My effort: Here $$\tan A=\frac{1-\cos B}{\sin B}$$ Now $$\begin{align}\text{L.H.S.} &=\tan 2A \\[4pt] &=\frac{2\tan ...
2
votes
1answer
63 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
0
votes
0answers
21 views

Proving that If $A+B+C=π$ then, [duplicate]

If $A+B+C=π$ then prove that : $\sin(B+2C)+\sin(C+2A)+\sin(A+2B)=4\sin\frac{B-C}{2}.\sin\frac{C-A}{2}.\sin\frac{A-B}{2}$. My attempts: Here $A+B+C=π$ Now, $$\begin{align} ...
1
vote
1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
1
vote
1answer
33 views

Showing that $\alpha$ satisfies the equation $\sin 2x=x$

This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme. Question: Here's what I have done: $$A(OBA) = \frac 12r^2α$$ [basic ...
0
votes
2answers
35 views

What is this procedure called for angle radians?

So, my lecturer says that $-\cos(\frac{\pi}{8}) = \cos(\frac{9\pi}{8})$. What did he do to get that? Please recommend a source where I can brush up on my knowledge of angles.