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

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Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...
0
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0answers
6 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
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2answers
33 views

Limit of 1/x - 1/(sin x) as x → 0

How to calculate this limit: $$\lim_{x\to0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$$ All I know is: $$\lim_{x\to0}(\sin x/x)=0$$ $$\lim_{x\to0}x=0$$ $$\lim_{x\to0}\sin x=0$$
3
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1answer
52 views

What does “versin” mean?

$$\newcommand{\versin}{\operatorname{versin}}2\versin A+\cos ^2 A= 1+\versin ^2 A$$ I don't understand the word 'ver' in this equation. What does it mean?
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1answer
10 views

how to find out intial direction and angle of collision

i have a problem in my game. I have a wall where a ball hit to a wall from anywhere. i need to give it to the direction according to collision law. Let suppose if a ball thrown from x == 0 and y == 0 ...
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0answers
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Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
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1answer
24 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
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0answers
22 views

Pipe measurements calculated wrong, can we fix it mathematically? [on hold]

While using a tool to calculate the ovality of a pipe, we had our tool calibrated wrongly and ended up with a spreadsheet of wrong values. The tool is normally placed in the center of the pipe and ...
0
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1answer
32 views

Integrate $dx/(4x^2-1)^{3/2}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
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2answers
47 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
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3answers
87 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
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4answers
35 views

Equivalence of equations

$ \sin ^2 \alpha = \frac{\tan ^2 \alpha}{1+\tan^2 \alpha} $ $ 1+\tan^2 \alpha = \frac{\tan ^2 \alpha}{\sin ^2 \alpha} $ It is said that these two equations are equivalent. How can that be? I know ...
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6answers
97 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
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4answers
50 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
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0answers
26 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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1answer
20 views

The vertical projection of a chord of a circle?

I was wondering if anyone could help me with the problem below (finding x): So we are given t_i (the initial tangent angle to the circle), t_o (the exiting angle of the tangent of the circle), the ...
3
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1answer
18 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
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3answers
43 views

What is the required radius of the smaller circles around a larger circle so they touch?

I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box) I would like to be able to adjust the number of ...
3
votes
3answers
48 views

Using trig substitution to solve for integration?

So I used a trig sub for this problem: $$\int \frac{1}{x^2\sqrt{9-x^2}}dx.$$ ${x=3\sin\theta}$ ${dx=3\cos\theta\ d\theta}$ ${\sqrt{9-x^2}= 3\cos\theta}$ I ended up with $$\frac19 \int \frac{ ...
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2answers
66 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
3
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1answer
40 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
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1answer
31 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
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0answers
23 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
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0answers
61 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
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2answers
49 views

How to solve $\sin(\arctan((\frac{1}{2}))$ [on hold]

Can you solve $\sin(\arctan((\frac{1}{2}))$? It says I have to use a right triangle
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4answers
55 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
3
votes
4answers
103 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
3
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2answers
97 views

Simple Equation Does my proof work?

Its the inequality equation $|a+b| \leq |a|+|b | $ I managed this by cases. Let $c = a$ and $d=b$ if $a>b $ let $c = b$ and $d = a$ if $b>a $ if $a=b$ let $a=c$ Hence we have $|c+d| \leq ...
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0answers
85 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-2mx+N = ...
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2answers
94 views

Prove that $\lim_{x\to\frac{2}{\pi}}\big\lfloor\sin\frac{1}{x}\big\rfloor=0$ [on hold]

Prove that $$\lim_{\large x\to \frac{2}{\pi}} \left\lfloor\sin\left(\frac{1}{x}\right)\right\rfloor=0$$ using the $\varepsilon$-$\delta$ definition of limits. Note that $\lfloor 0.1\rfloor = 0,\; ...
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2answers
51 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...
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1answer
24 views

Isosceles has maximum vertex angle between triangles of equal area

I'm trying to prove the following that in the image below (E1 & E2 are parallel, AB=AC) no matter where I move the vertex point A on line E1 (keeping BC as is), the vertex angle A is going to ...
2
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2answers
54 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
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0answers
26 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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2answers
39 views

Sine on a Circle

I'm walking a quarter mile circular walking track. The width of the track is 8 feet across. If I walk from one side of the track to the other, walking a sine wave that has a 20 foot period, how much ...
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0answers
31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
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0answers
86 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
2
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2answers
53 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
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0answers
20 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
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3answers
183 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
0
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1answer
42 views

How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?

Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$ Now, ...
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4answers
78 views

prove that $\sqrt{2} \sin10^\circ+ \sqrt{3} \cos35^\circ= \sin55^\circ+ 2\cos65^\circ$

Question: Prove that: $\sqrt{2} \sin10^\circ + \sqrt{3} \cos35^\circ = \sin55^\circ + 2\cos65^\circ$ My Efforts: $$2[\frac{1}{\sqrt{2}}\sin10] + 2[\frac{\sqrt{3}}{2}\cos35]$$ $$= 2[\cos45 \sin10] ...
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3answers
35 views

trigonometric identity of sin squared in terms of tan squared.

Why is $\sin^2(x)=\frac{\tan^2(x)}{1+\tan^2(x)}$? And why is $\sin^2(x)=\frac{1}{\cot^2(x)}$? I've tried starting from $\tan^2(x)=\frac{\sin^2(x)}{1-\sin^2(x)}$ but that wasn't really working out ...
0
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1answer
25 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
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0answers
39 views

can somebody help me with this? [on hold]

Just gonna ask you guys if it's possible to prove that $\sec A\tan A - \sin A\sec A = 1 - \tan A$
2
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1answer
36 views

Finding the third side of a triangle, given ratio of two sides and difference of two angles [on hold]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
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1answer
34 views
2
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2answers
60 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
3
votes
3answers
86 views

Simplify a quick sum of sines [duplicate]

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$ I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.
1
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4answers
79 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?