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

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17
votes
5answers
2k views

A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
2
votes
2answers
65 views

How to easily solve this trigonometric equation?

Given equation: $$\frac{\sin(x) + \sin(5x) - \sin(3x)}{\cos(x) + \cos(5x) - \cos(3x)} = \tan(3x),$$ what is the easiest way to solve it? I know it can be solved by expanding each $\sin(nx)$ and $\cos(...
0
votes
2answers
51 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
0
votes
1answer
77 views

Collinearity of triangle vertices on circular paths

Suppose as a function of time, the vertices of a triangle move with constant speed along circular trajectories: $$ \vec p_i(t) = a_i \left( \begin{array}{c} \cos(b_i t+ c_i)\\ \sin(b_i t+ c_i) \end{...
1
vote
0answers
23 views

Small circles on sphere: finding angles for constant “cosine” onto a parallel.

My problem can be best explained starting from a 2D example: Imagine having a circle and wanting to discretize N points on the circumference of the circle so that the difference of the cosine of each ...
3
votes
3answers
74 views

Is there a mathematical reason why rotation in the counterclockwise direction positive and clockwise rotation negative?

This inquiry has recently come to me in my study of trigonometry and the unit circle. It was said right from the very start that counterclockwise rotation were positive while clockwise rotations are ...
0
votes
0answers
11 views

How are extracted this formulas from Tait-Bryan rotation matrix

I've a Tait-Bryan rotational matrix with this order X,Y,Z: Matrix Image I want to extract a Roll angle and Pitch angle from this matrix, and this are the two formulas: Matrix Equations Image I ...
-1
votes
1answer
20 views

how to get the angle of arc ??

dart game board is divided into sectors by 30 degrees like pizza slice. the given is (x, y) coordinates, and I need to find where coordinates are lying on. how can I get the angle just with ...
-1
votes
0answers
46 views

How to do this problem [on hold]

Given: $$\sin(\theta')+\sin(\theta")+\sin(\theta''')=3$$ Find: $$\cos(\theta')+\cos(\theta")+\cos(\theta''')=?$$ How to solve this equation i tried to convert every term in terms of cos but it ...
0
votes
0answers
23 views

Shortest route around a circle [on hold]

Sorry I am very short of sleep so I hope this makes sense. I have a revolving stage which is programmed to move in a sort of linear fashion. One rotation of the revolve CW from a zero point would be ...
3
votes
2answers
124 views

If the sides of a triangle satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$, and if one angle is $48^\circ$, then find the other angles.

In triangle $ABC$ one angle of which is $48^{\circ}$, length of the sides satisfy the equality: $$(a-c)(a+c)^2+bc(a+c)=ab^2$$ Find the value in degrees the other two angles of the triangle. I ...
2
votes
1answer
55 views

Brocard Angles proof by Sine and cosine formulae.

The angles denoted by $\omega$ are the Brocard angles. Recently i came to know about the Brocard Angles and also their property i.e $\cot{\omega}=\cot{A}+\cot{B}+\cot{C}$. In my previous question I ...
-2
votes
1answer
39 views

What is the value of tan θ in this problem?

If $\vec{A} = 4\vec{i}+3\vec{j}$, $\vec{B} = 5\vec{i}-12\vec{j}$ and $\theta$ is the measure of the angle between the two vectors $\vec{A}$ and $\vec{B}$, then what is the value of $\tan \theta$?
0
votes
0answers
19 views

Find the graph of the given function

$f(x)=$$Sin^{-1}$${(3x-4x^3)}$ , Plot the graph for $f(x)$ I want to know how I can plot graph from this points.
3
votes
6answers
132 views

If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$then..

If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$ then find the value of $f(0) + f'(0) + f''(0)$. I tried differentiating the given. But it is getting too long and ...
1
vote
2answers
81 views

Prove that: $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$

Let $n\in \mathbb{N} , n> 1$ Prove that : $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$ I know: $f(x) = \tan x$ is convex function $f(a x + b y) < a f(x) + b f(y), a+b=...
1
vote
2answers
51 views

If $\sin(\pi \cos\theta) = \cos(\pi\sin\theta)$, then show …

If $\sin(\pi\cos\theta) = \cos(\pi\sin\theta)$, then show that $\sin2\theta = \pm 3/4$. I can do it simply by equating $\pi - \pi\cos\theta$ to $\pi\sin\theta$, but that would be technically wrong as ...
0
votes
4answers
105 views

Value of $\int\tan^{-1}(x)\,dx$

What is the value of $\int^{1000}_{0}\tan^{-1}(x)\,\mathrm d x$? Today we were taught about graphs of all trigonometric inverse functions. So my proofessor split it into $0-\tan(1)$ and $\tan(1)-...
1
vote
1answer
64 views

If xy + yz + zx = 1, …

If $xy + yz + zx = 1$, then show that $$\dfrac{x}{1-x^2} + \dfrac{y}{1-y^2} + \dfrac{z}{1-z^2} = \dfrac{4xyz}{(1-x^2)(1-y^2)(1-z^2)}$$ I tried doing the sum algebraically, that is, by solving ...
0
votes
4answers
119 views

Strange integral result

Consider the following integral, $$\mathrm{I} = \int_{-1}^{1}\frac{d}{dx}\tan^{-1}\left(\frac{1}{x}\right)dx$$ We can do this in two ways, First Using the fact that the antiderivative of $\frac{d}{...
1
vote
0answers
92 views

Too obvious to rigorously prove?

Problem Prove that $\pi$ is a half-period of the function $y=\cos x$. Is $\pi$ a half-period of the function $y=\tan x$? of $y=\cot x$? Solution Attempt We know the period of $\cos x$ is $2\...
1
vote
5answers
96 views

With the linear approx. of $f(x)= sin(x)$ around $0$ Calculate $\lim_{\theta\to 0} \frac{sin\theta}{\theta}$

With the linear approximation of $f(x)= sin(x)$ around $0$, calculate: $$ \lim_{\theta\to 0} \frac{\sin\theta}{\theta}$$ Figured I have to use L'Hospital's Rule, but I think I don't get how to ...
1
vote
2answers
45 views

I am trying to prove that, ${\sin{A}\cos{A}-\sin{B}\cos{B}\over \sin^2{A}-\sin^2{B}}=\tan(90^o-A-B)$

I am trying to prove that, $${\sin{A}\cos{A}-\sin{B}\cos{B}\over \sin^2{A}-\sin^2{B}}=\tan(90^o-A-B)$$ Using: $\sin(2A)=2\sin{A}\cos{A}$, then we have $${1\over 2}{\sin(2A)-\sin(2B)\over \sin^2{A}-\...
2
votes
0answers
67 views

Question about an infinite product

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
0
votes
2answers
21 views

Expanding double angles and simplifying trigonometry

guys I am struggling with the expansion from the second last line to the last for this problem. Any suggestions with examples would be highly appreciated. Thanks a lot
1
vote
0answers
27 views

General solution to an trigonometric equation set

An extent for last question I asked is to solve $$\begin{cases} 2\cos(2y_1)=1+\cos(y_1+y_2)\\ 2\cos(2y_2)=\cos(y_1+y_2)+\cos(y_2+y_3)\\ 2\cos(2y_3)=\cos(y_2+y_3)+\cos(y_3+y_4)\\ ...\\ 2\cos(2y_{n-1})=\...
-2
votes
3answers
55 views

Show that $\sin 3 \theta = 3\cos^2\theta \sin\theta - \sin^3\theta$ [on hold]

Use the de Moivre's formula to derive the following trigonometric identities $$\sin 3 \theta = 3\cos^2\theta \sin\theta - \sin^3\theta$$
1
vote
1answer
23 views

Find intersection angle of curves : $y=x^3-5, y=5x^2-x$

My work so far looks as above. Calculated angle seems too large? Not sure what's wrong in here with my calculations. Thanks!
-1
votes
0answers
11 views

Change in azimuth of target after I execute a roll [closed]

Given the azimuth of an object from an aircraft, x°, what would be the new azimuth of the object if the aircraft executes a roll of y°?
0
votes
1answer
56 views

Given the bounds of a rotated ellipse, can you find the semi-major and semi-minor axis?

Clarification: I am trying to find the semi-axes $(a,b)$ given the bounding rectangle's dimensions $(x,y)$. To constrain the problem, I am keeping $\theta$ the same as my original ellipse. The ...
2
votes
3answers
137 views

If $\sin\theta + \sin\phi = a$ and $\cos\theta + \cos\phi = b$, then $\sin(\theta+\phi) = ???$ [closed]

If $$\begin{align} \sin\theta + \sin\phi &= a \\ \cos\theta + \cos\phi &= b \end{align}$$ then find the value of $\sin(\theta + \phi)$. How to solve? Please. help.
-2
votes
0answers
43 views

Simple trigonometric division question

studying for a physics exam and a bit rusty on trigonometry having not done it for four years. I am following the answer to a friction of a car on a banked curve question and I am fine before and ...
0
votes
1answer
45 views

How do we know how many special right triangles there are?

Note: Not a duplicate of this. So the so-called "special right triangles", or 30-60-90 and 45-45-90, are special triangles that have sines, cosines and tangents that can be calculated easily. Are ...
0
votes
1answer
32 views

How to ensure to find all solutions of a trigonometric equation?

Hi, so I was doing this question and I found the solution $\theta = \frac{5}{12}\pi$ equating $\theta+\frac{\pi}6$ to $\pi-\theta$. However, there is also another solution for this equation $\frac{11}{...
0
votes
1answer
25 views

How many solutions of any given triangle (SSA) are there using the sine law?

How can you tell how many solutions a triangle will have, given SSA? In addition, how can each solution be verified?
1
vote
2answers
51 views

arcTangent without a calculator, power series is not fully capable

Right now I'm using Java code to have a turret rotate to track a target. In order to get the angle of rotation I'm using a power series from n = 0 to 100 to get a close estimate ($\sum_{n=0}^{100} \...
2
votes
0answers
46 views

An infinite product for $\arccos \alpha$ similar to Viete's formula for $\pi$

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
0
votes
1answer
25 views

Distance Between Two Points on the Surface of a Cone

A right circular cone has radius 3 and height 3. If A and B are two points on its surface, what is the maximum possible straight-line distance between A and B? I fixed $A$ to be a point on the ...
0
votes
6answers
89 views

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$?

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$ ? All I could till now : LHS =$2\sin{x}+\cos{x}$ Since, $−\sqrt{5} \leq 2\sin{x}+\cos{x} \leq \sqrt{5}$ So a ...
-4
votes
0answers
52 views

triangle and a circle where the sum of triangle angles is $pi$ [closed]

Assuming there are three angles and a circle is the sum of three angles in trigonometry ,is the triangle the only geometry figure where the sum of the angles gives $pi$ and is it different from all ...
2
votes
2answers
45 views

Solve Sum of Arccos

I am working through a situation with trying to fit an equilateral triangle into a square, and I have boiled it down to the following equation: $$\arccos\left(\frac{\frac{L}{2}+x}{S}\right) + \arccos\...
3
votes
4answers
149 views

Different ways of evaluating $\int_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx$

My friend showed me this integral (and a neat way he evaluated it) and I am interested in seeing a few ways of evaluating it, especially if they are "often" used tricks. I can't quite recall his way,...
-2
votes
3answers
49 views

Help with Trigonometric Functions

so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the sin, cos and tan functions, I am not posting the formula for obvious reasons. My ...
0
votes
1answer
37 views

Finding the length of a triangle using Sine Law

I am having trouble solving, for this triangle. I am trying to find RS using the Sine Law So, $${a \over Sin A}= {b \over Sin B} \\ {25.6 \over Sin 120} = {b\over Sin 28} \\ {25.6 \over Sin 120} \...
0
votes
2answers
90 views

How to prove $(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$

Q: How to prove $$(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint.
4
votes
4answers
73 views

Proving $\frac {\sin x}{1-\sin x}-\frac {\sin x}{1+\sin x}\equiv 2\tan^2 x$

I need assistance with proving the following identity: $$\frac {\sin x}{1-\sin x}-\frac{\sin x}{1+\sin x} \equiv 2\tan^2 x$$ What I have done so far is expanded them: $$\frac {\sin x\;(1+\sin ...
3
votes
2answers
124 views

Is this notation mathematically-correct? $\cot\alpha\pm\tan\alpha=\frac2{{\sin\atop\tan}(2\alpha)}$

I have a question. Look at the following expression: $$\cot\alpha\pm\tan\alpha=\frac2{{\sin\atop\tan}(2\alpha)}$$ Is it written well, according to the laws of mathematical language? In that ...
0
votes
1answer
54 views

Barbell in a Room [closed]

A 7' x 7' room contains a 7' barbell placed in the center of the room, parallel to the north and south walls. How many degrees must the barbell be turned in order to provide exactly 1' of separation ...
-2
votes
0answers
25 views

Addition and multiplication of trigonometric series [closed]

Sir, which part of trigonometric this part comes? Is it available in KC Sinha?