Tagged Questions

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

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0
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0answers
14 views

Solving spherical triangle

How do you use Napier's analogies to find the angles $\alpha$ and $\beta$ in here ?
4
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1answer
38 views

Why do we have trigonometric functions besides $\sin(x)$?

Probably a terrible question, but I've been curious and can't come up with a reason besides convenience for myself with my limited knowledge. Why do we have $\cos(x)$, $\tan(x)$, etc. when all of ...
-3
votes
2answers
29 views

Integrate using trigonometric substitutions: [on hold]

Integrate $\frac {\sqrt{4x^2+4}}{x} $ using trigonometric substitutions
-6
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3answers
32 views

Integrate $\frac{x^3}{(1-x^2)^{1/2}}$ using trigonometric substitition [on hold]

Use trigonometric substitition to integrate $$\int\dfrac{x^3}{(1-x^2)^{1/2}}\,dx$$
3
votes
3answers
58 views

Solve: $\sin x - y\cos x = z$ for $x$.

I am working on programming a series of algorithms into a project, however I have run into trouble trying to solve this equation for $x$: $$ \sin x - y\cos x = z $$ It should be noted that $y$ and ...
0
votes
0answers
6 views

Circumcentre of three points X, Y, Z, given distance from each to points A and B

I'm racking my brain trying to figure out where to start on this, and it's been too many years since working on these kinds of problems. I have six measurements which I'd like to use to calculate a ...
1
vote
2answers
21 views

Can known object be used to back-calculate my location?

I apologize if this is in the wrong forum. Wasn't sure to put it here or navigation. Say I have a map, and on it, I know the range and bearing/heading of a known object. Is it possible to somehow ...
1
vote
4answers
48 views

Solving this trigonometric task

Find the values of $R$ and $\alpha$ in the identities below, given that $R>0$ and $\alpha$ is an acute angle. $$\sqrt{3}\cos{\theta}-\sin{\theta}=R\cos(\theta+\alpha)$$ I'm a bit confused by this ...
1
vote
3answers
40 views

Finding the limit of a function with ArcTan

I've found difficulties finding this limit (without using Taylor series approximation, as it's intended for the secondary-school): $$\lim_{x\to \infty} {x^3\over(x^2+1)\arctan(x)}-{2x\over \pi}$$ ...
4
votes
2answers
57 views

Solving $\sin(2v) = \sin(v)$

$$\sin(2v) = \sin(v)$$ Why can't this equation be solved by setting: $$2v = v + 2\pi n \quad \leftrightarrow \quad v = 2\pi n\\2v = \pi - v + 2\pi n \quad\leftrightarrow \quad 3v = \pi + 2\pi n ...
0
votes
1answer
36 views

Beautiful problem about an ellipse and its eccentricity

If the tangent at a point (a cosθ,b sinθ) on the ellipse meets the auxiliary circle in two points, the chord joining them subtends a right angle at the centre, then the eccentricity of the ellipse is ...
2
votes
2answers
33 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $ [on hold]

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
2
votes
1answer
56 views

Beautifully looking little geometry/trigonometry problem

Given triangle ABC, a,b,c as its sides, p is a half perimeter, such that $\dfrac{p-a}{11}=\dfrac{p-b}{12}=\dfrac{p-c}{13}$. We need to find $(\tan\dfrac{A}{2})^2$ (A)$\dfrac{143}{432}$ ...
3
votes
5answers
98 views

What is the limit of this trig function?

How do I find $$\lim_{x \to \pi/4}{\frac{\cos x-\frac{1}{\sqrt2}}{x-\frac\pi4}}$$? I've tried setting the denominator equal to $h$, then replacing $x$ in terms of $h$, but I still don't know how to ...
0
votes
1answer
42 views

power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
1
vote
3answers
48 views

Find the angle between hour hand and minute hand at 1:59?

I have got the formula to find angle between hour hand and minute hand from http://en.wikipedia.org/wiki/Clock_angle_problem The angle between the hands can be found using the formula: ...
0
votes
1answer
15 views

Solving for joint angles in 2-segment robot leg

I am trying to program a robot leg with 2 segments and two joints, such that for a given location of the foot, I can calculate the angles of both joints. From here on out, the positive Y direction is ...
-1
votes
3answers
47 views

Express $\cos \left ( 5x \right )$ via powers of $\sin \left ( x \right )$ and $\cos \left ( x \right )$?

Using De Moivres formula and Newtons binomial theorem. Also, express $\cos ^{5}\left ( x \right )$ via trigonometric functions of multiple angles. What I've managed to do so far: ...
11
votes
4answers
156 views

Ways to prove $\displaystyle \int_0^\pi dx \dfrac{\sin^2(n x)}{\sin^2 x} = n\pi$

In how many ways can we prove the following theorem? $$I(n):= \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$$ Here $n$ is a nonnegative integer. The proof I found is by considering ...
1
vote
2answers
82 views

Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$

Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$ for any $x\in\mathbb{R}\setminus{k\pi}$ where $k\in\mathbb{Z}$. I wrote $\sin5x$ as $5\cos^4x\sin{x}-10\cos^2 x\sin^3x+\sin^5x$ and ...
3
votes
0answers
22 views

Evaluating $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$ without Complex Numbers [duplicate]

Alright, so the standard way to evaluate $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$, is to respectively take the real and imaginary part of $$\sum_{k=0}^n{\rm e}^{ikx}={\frac ...
0
votes
2answers
25 views

Pls Help Explain This Confusing Explaination for sin(a)=sin(180​∘​​−α) for any angle α

Can anybody help explain why sin(Z)=sin(θ) in the image that I provided below? *I put the confusing part in red rectangle. It's clear in the diagram that sin(Z), i.e. ∠XZY, is bigger than sin(θ) and ...
1
vote
3answers
32 views

Is my working out for Tan's equivalent correct?

So if i have: $$\tan(\frac{\pi}{2}+\theta)$$ Am i able to: $$\frac{\sin (\frac{\pi}{2}+\theta)}{\cos(\frac{\pi}{2}+\theta)}$$ $$=\frac {-\sin\theta}{\cos\theta} = -\tan\theta$$ Or am i ...
3
votes
1answer
46 views

Formula for Determinant of Vectors given in spherical coordinates

In 2D, one has an easy formula for the determinant of two vectors given in spherical coordinates, i.e. $\begin{vmatrix} \cos(\phi_1) &\cos(\phi_2)\\ \sin(\phi_1) &\sin(\phi_2)\end{vmatrix} ...
0
votes
2answers
18 views

The range of arc-cotangent function & arccot(-1).

We know that the range of arc-cotangent function is $(0,π)$ and we I calculate the value of $cot^{-1}(-1)$ by a calculator, I get ($-π/4$) Which is clearly not included in the range !! Why isn't it ...
8
votes
3answers
182 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$\int_0^1 (\arctan x)^2 \ln\left(\frac{1+x^2}{2x^2}\right) {\rm d}x$$ I substituted $x=\tan\theta$ and got $$-\int^\frac{\pi}{4}_0 \theta^2\frac{\ln(2\sin^2\theta)}{\cos^2\theta} ...
0
votes
1answer
21 views

Two roots of $\arcsin(x)$ in the range $[0,2 \pi]$

I am baffled with how to write the two roots of arcSin$(x)$ in the range $[0,2 \pi]$, while $x \in [-1,1]$, such that one root can be directly calculated in terms of the other root. For instance, we ...
7
votes
0answers
109 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
0
votes
2answers
63 views

What does adding $\sin\theta \cos\theta$ make my graph a linear relationship?

What is the point of adding sin n cos of theta when graphing range? e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationship. ...
1
vote
1answer
20 views

Find zeroes of trigonometric polynomial

I know this is a rudimentary question but I'm not really sure how to do this. For my homework problem I have to verify some error term of trapazoidal quadrature. I end up with $$f^{(3)} = -8\sin ...
0
votes
2answers
23 views

Solving triangles with trig, word problem

Engineers want to measure the distance from P to Q, but the span from P to Q is across the tip of a lake. So they select a point R on land and find that the distance from R to Q is 100 feet and from R ...
2
votes
1answer
29 views

Trigonmetric sum of inverses

Prove that: $$\sum^{45}_{k=1}\frac{1}{\cos1^\circ-\cos(87+4k)^\circ}=\frac{1}{2\sin 1^\circ}$$ Numerically, this is accurate comparing the lhs and rhs. Some ideas: We can transform the question ...
0
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0answers
9 views

how to add statute miles to latitude and longitude coordinates in decimal notation

After some reading I've got questions. If 1 nautical mile = roughly 1.1508 statute miles, and I have a latitude of 26.8230556 (decimal notation), can I simply add/subtract a constant decimal that ...
0
votes
1answer
18 views

How can this equation be simplified this way? Transmission line: Zin

I thought of putting this on the Electrical Engineering Exchange but I thought since this seems more mathematical than related to engineering I thought I should place it here instead. Question: Why ...
-2
votes
2answers
30 views

Solving Trig Equations Using Identities [on hold]

$\sin^2 x = \cos^2 x + 1$ Solve for $x$ in radians Any help would be much appreciated
5
votes
2answers
120 views

Evaluation of $-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $

How would I integrate this: $$-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $$ I have tried several methods but can't seem to work this out.
1
vote
0answers
41 views

How to prepare myself for an advanced trignonometry exam

I'm gonna have a trigonometry/general algebra exam soon. My teacher has told us about some trignometric proofs, and we defined the $\sin$ and $\cos$ int he right way, doing all formal proofs for the ...
0
votes
2answers
31 views

How Do We Find Points On A Circle Equidistant from each other?

I'm a programmer and I saw this question on stackoverflow which exactly does my job: http://stackoverflow.com/questions/13608186/trying-to-plot-coordinates-around-the-edge-of-a-circle. In this, the ...
-1
votes
0answers
44 views

Show that $\frac{\tan x + \cos y }{ \tan x \cos y} = \tan y \cos x $

Show that $$\frac{\tan x + \cos y }{ \tan x \cos y} = \tan y \cos x $$ I was given this problem due to me losing to my teacher in a game of cards and he expects it done tomorrow but honestly ...
0
votes
2answers
29 views

Trisection of an angle with straightedge and a compass

Suppose there exists an angle Z such that cos Z = -11\16 Prove or disprove that such an angle can be trisected with a straightedge and a compass. Well, we know that an angle is constructible iff its ...
0
votes
1answer
26 views

Why didnt it fit? [on hold]

Grandmother Stewart started a family quilt years ago. As each family member becomes suffciently skilled, he or she is given the task of designing and completeing a new square for the quilt. IT has ...
0
votes
0answers
35 views

Trig and graphing

I am stuck and do not know how to do this problem; list all points on the graph of $y=tan(x)$ on the interval $[\frac {\pi} {2},3 \pi]$ that have a $y$ coordinate of $\frac {-1} {\sqrt 3}$
0
votes
1answer
28 views

Time based algorithm to make object orbit another in electron-type path?

I'm positioning an object in 3d space, and I want to make it orbit another object, in a semi-random electron-like orbit, such that it always stays the same distance from the origin. I can make it move ...
-1
votes
0answers
42 views

Trigonometry : Application [on hold]

Let 0◦ ≤ α, β, γ ≤ 90◦ be angles such that sin α − cos β = tan γ sin β − cos α = cot γ What is the sum of all possible values of γ in degrees.
0
votes
2answers
29 views

Need help finding algorithm to fix specified problem

First I want to say that I am not a mathematician, so asking a question in this area is not easy for me. So I will describe the issue in my words which is not the nice way. So this is what I do: I ...
2
votes
0answers
30 views

Trigonometry equation, always ending up with root of 17

how would one approach to solve this equation? $4sin^2x - ctg^2x = 0$ I transform it into a quadratic equation in which t = cosx, however i keep ending up with the wrong result and i cant seem to get ...
2
votes
2answers
44 views

Why is Implicit Differentiation needed for Derivative of y = arcsin (2x+1)?

my function is: $y = arcsin (2x+1)$ and I want to find its derivative. My approach was to apply the chain rule: ${y}' = \frac{dg}{du} \frac{du}{dx}$ with $g = arcsin(u)$ and $u = 2x+1$. ${g}' = ...
1
vote
2answers
29 views

$\sin(x+\pi/3)=2\sin x \sin(\pi/3)$ show that $11\tan x=a+b\sqrt{3} $ ; $a,b$ are elements of positive integers

If $x$ satisfies the equation $\sin\left(x+\dfrac{\pi}{3}\right)=2\sin x\sin\left(\dfrac{\pi}{3}\right)$, show that $11\tan x=a+b\sqrt{3}$, where $a,b\in\mathbb{Z}^+.$ [src] I've tried using ...
4
votes
3answers
51 views

Verifying $\sec^2x + \tan^2x = (1-\sin^4x)\sec^4x$

Verify: $$\sec^2x + \tan^2x = (1-\sin^4x)\sec^4x$$ My solution: $$ \begin{align}\sec^2x+\tan^2x&=\frac{1}{\cos^2x}+\frac{\sin^2x}{\cos^2x}\\ &=\frac{1+\sin^2x}{\cos^2x}\\ ...
0
votes
3answers
18 views

Obtaining normal form of a line from the general form

This is a question relating to SL Loney's coordinate geometry book (article 56). We have $Ax + By + C = 0$ as the general form of a line. Want to arrive at $xcos(\alpha)+ ysin(\alpha) - p = 0$ as ...