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

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4
votes
1answer
97 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
0
votes
2answers
30 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
1
vote
4answers
73 views

Solutions of $\sin^2\theta = \frac{x^2+y^2}{2xy} $ [closed]

If $x$ and $y$ are real, then the equation $$\sin^2\theta = \frac{x^2+y^2}{2xy}$$ has a solution: for all $x$ and $y$ for no $x$ and $y$ only when $x \neq y \neq 0$ only when $x = y \neq 0$
0
votes
1answer
36 views

Prove that $-4\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq10$

Prove that $$\color\red{-4}\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq\color\red{10}$$ My attempt:- I simplified the equation to $$\begin{align} &\;\;\phantom{=} 5\cos\theta+3\cos(\...
-1
votes
1answer
55 views

Find the value of x of $\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? [closed]

$\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? How to find the value of $x$ I've been thinking for this question for quite a time. Hope can somebody solve it. ...
0
votes
0answers
34 views

How to solve this $80^\circ$-$80^\circ$-$20^\circ$ triangle ($60^\circ+20^\circ$ and $70^\circ+10^\circ$ variant)? [duplicate]

A friend of mine asked me for help with a math problem and I struggled with this for over an hour. I told him sorry, and I felt bad. It's been bugging me now for hours. I don't even so much care for ...
1
vote
2answers
50 views

Why should the solutions of $(\sin x)^2 = 0$ be rejected in the equation $((\sin x)^2)(\csc x + 1) = 0$?

Q: Determine the number of solutions for $((\sin x)^2)(\csc x + 1) = 0$ over the interval $0 \leq x < 2\pi$ with the correct reasoning. Correct answer: There is one solution because the solutions ...
2
votes
1answer
41 views

Prove that $\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$ using given condition [duplicate]

If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$ I tried using the fact that $\...
5
votes
2answers
108 views

Roots of $y=x^3+x^2-6x-7$

I'm wondering if there is a mathematical way of finding the roots of $y=x^3+x^2-6x-7$? Supposedly, the roots are $2\cos\left(\frac {4\pi}{19}\right)+2\cos\left(\frac {6\pi}{19}\right)+2\cos\left(\...
-3
votes
0answers
25 views

Is there a way to measure an angle, without using any protractor or trig finctions that requires a calculator? [closed]

Say i have an angle printed on a paper, Is there a way to measure it?
-1
votes
0answers
64 views

How to derive two angles and a length from this diagram [closed]

I'm familiar with sohcahtoa, the sin and cosine rule, I just can't seem to apply them here. I know angles alpha and ow, I know lengths z and al. I need to know length ? and angle ?? and angle Q
2
votes
2answers
40 views

Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$

According to my book $$\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ when $x<0$, $y<0$, and $xy>1$. I can't understand one thing out here that when the above ...
0
votes
3answers
42 views

$\tan x= \cot (x+\phi)$ for some $\phi$

Suppose we graphed the equation $ y = \tan x $. Is it possible to describe this graph with an equation of the form $ y = \cot (x + \phi) $, for some number $ \phi $? Why or why not?
5
votes
4answers
114 views

Tips for integrating $\int \frac{dx}{1+\cos(x)}$

I tried the following $$ \int \frac{dx}{1+\cos(x)}=\int \frac{1-\cos(x)}{1-\cos^2(x)}\,dx= \int \frac{1-\cos(x)}{\sin^2(x)}\,dx\\ =\int \frac{1}{\sin^2(x)}\,dx-\int \frac{\cos(x)}{\sin^2(x)}\,dx=\int ...
3
votes
2answers
39 views

Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
1
vote
1answer
25 views

Inverse trigonometry simple doubt.

I recently started learning the inverse trigonometric function and got stuck at one point. In the question involving the expression of tangents, It was given that $\frac{x}{y}>1$ The authors ...
0
votes
4answers
49 views

Small problem about domain of a function .

I want to know that whether $f:\mathbb{R}^2/\lbrace(0,0)\rbrace \to \mathbb{R}$ defined by $f(x,y) = \arctan(\frac{x}{y})$ is a function or not? I think this is very silly problem but i think it is ...
0
votes
1answer
83 views

Why are trigonometric functions defined so abruptly?

I will make use of a diagram to ask the question I am asking. First of all, consider this diagram: This will be enough for me to ask a question. Now, let $AC=AB=1$, which means both hypotenuse are ...
9
votes
1answer
100 views

Fake proof that $\frac{e^x-1}{e^x+1}=e^x$, via integrating $\operatorname{sech} x$ in two ways

We start with the integral: $$\int \text{sech}(x)dx$$ Method 1 \begin{align} \int \text{sech}(x)dx & = \int\frac{2}{e^x+e^{-x}}dx \\ &= \int\frac{2e^x}{e^{2x}+1}dx \end{align} Using the ...
1
vote
2answers
34 views

If $a\sin x + b\cos(x+\theta) +b\cos(x-\theta) = d$, then what is the minimum value of $|\cos\theta|$?

If $$a \sin x + b \cos(x+\theta)+ b \cos(x-\theta) = d$$ then what is the minimum value of $|\cos\theta|$? The answer is given: $\dfrac{\sqrt{d^2 - a^2}}{2|b|}$ I tried simplifying the ...
3
votes
1answer
67 views

Find real parts of the complex roots of this $9^{th}$ order polynomial in explicit form

I have a following polynomial. (See WolframAlpha ): $$x^9-6x^8+14x^7-16x^6+36x^5-56x^4+ 24x^3-320x+\frac{640}{9}=0 \tag{1}$$ Wolfram says that $(1)$ has three real roots and three pairs of complex ...
0
votes
5answers
80 views

How do you solve trig integrals using recursions?

My calculus professor gave us the problem $\int \sin^2(x)\cos^2(x)dx$ and told us to solve it via recursion but I can't seem to find how to do it in my textbook.
0
votes
1answer
32 views

Generalize $r\cos(\theta n)$ Into Polynomials in Terms of x

I understand that it is possible to generalize $\cos(\theta n)$ via Chebyshev polynomials of the first kind, and I was also wondering if it is possible to generalize $r\cos(\theta n)$ in a similar ...
3
votes
0answers
91 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
1
vote
1answer
34 views

How to calculate $\lim_{x\to4}(x-4)\cdot\cot(x-4)$

It's been a while since I've done calc, so I'm trying to review by reading "Calculus Demystified" by Steven G. Krantz. Question 1c at the end of chapter 2 has me stumped: $$\lim_{x\to4}(x-4)\cdot\cot(...
3
votes
1answer
75 views

I want to show that, ${\Phi\tan{9^\circ}-\phi\tan{27^\circ}\over \sin^2{9^\circ}-\sin^2{27^\circ}}=4$

$\phi$: golden ratio, $\Phi={1\over \phi}$ I want to show that, $${\Phi\tan{9^\circ}-\phi\tan{27^\circ}\over \sin^2{9^\circ}-\sin^2{27^\circ}}=4$$ Using $\sin^2{x}={1\over 2}(1-\cos{2x})$ $${\Phi\...
0
votes
1answer
37 views

Eliminate $x$ from this system: $a \sec{ x} +b\tan{ x} +c = 0 $ and $ p \sec {x} +q\tan {x} + r = 0$

Eliminate x from this system of equations: $$\begin{align} a \sec{ x} +b\tan{ x} + c &= 0\\ p \sec {x} +q\tan {x} + r &= 0 \end{align}$$ I tried expressing the value of the trigonometric ...
1
vote
6answers
183 views

Solve definite integral: $\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$

I need to solve: $$\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$$ Here is my steps, first of all consider just the indefinite integral: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...
1
vote
0answers
31 views

CFD: finding the vorticity magnitude the streamwise direction of an airfoil

I am doing CFD and I have to find the magnitude of the vorticity vector in the streamwise direction of an airfoil in every mesh cell. The streamwise direction is defined as being parallel to the ...
0
votes
2answers
56 views

What is the relation between $x,y$ if $\tan(20^\circ),x,\tan(50^\circ)$ and $\tan(20^\circ),y,\tan(70^\circ)$ are in AP?

If $\tan(20°),x,\tan(50°)$ are in AP and $\tan(20°),y,\tan(70°)$ are in AP then relation between x and y is?. $$\text{Attempt}$$. As they are in AP So $2x=\tan(20°)+\tan(50°),2y=\tan(20°)+\tan(70°)$ ...
19
votes
5answers
3k 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
54 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
81 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
24 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
75 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
13 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 [closed]

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 [closed]

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
127 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
60 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
133 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
82 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
65 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
126 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}{...