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

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1
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3answers
30 views

How to solve $\cos(5\alpha + \pi/2) = \cos(2\alpha + \pi/8)$ for $a$?

I missed the lecture. I don't want you to solve my homework, I just want to learn how to solve equations like this one. Since I have no idea, I'll post the task I got for homework, rather than ...
1
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1answer
32 views

Sum of trigonometric functions

Is the following inequality true? $$\left|\sum_{i=1}^{n}\left(\cos(x_i) \prod_{j\neq i}\sin(x_j)\right)\right|\le 1$$ I tried to count the extremes but it didn't work.
0
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0answers
16 views

When substituting in integration, do you have to change the limits of integration so long as you keep it consistent?

I have this integral: In order to solve for it, I have to substitute: t=tan(theta) dt=(sec(theta))^2 d(theta) When substituting that, I know I have to change the limits of integration within ...
1
vote
1answer
20 views

Find the radius given only a few variables

I'm writing a program that allows someone to generate a vertical road segment in 3D given a HEIGHT and an ANGLE. The road starts off flat, curves (to the ANGLE), has a brief straight segment (SEGLEN), ...
1
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4answers
51 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
2
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2answers
22 views

Inequalities with arctan

I don't understand how to solve inequalities with arctan, such as: $$\arctan\left(\frac{1}{x^2-1}\right)\ge \frac{\pi}{4} $$ If someone could solve this and give me a very brief explanation of what ...
2
votes
2answers
71 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
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2answers
39 views

Prove $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$

How do you prove that $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?
0
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1answer
46 views

Find exact value of $\sin\left(\dfrac x2\right) $

I have tried this problem over and over but can not get it. Can anyone provide a solution? Given $\sin(x) = -\dfrac67$ and $\tan(x)\gt0$ , find the exact value of $\sin\left(\dfrac x2\right) $.
3
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3answers
32 views

Verifying trig identities specific problem

$$\frac1{1-\cos y} + \frac1{1+\cos y} = 2\csc^2y $$ My attempt was me trying to find a common denominator on the left side but I don't know what to do after that.
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2answers
990 views

Proof that $\sin 10^\circ$ is irrational

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin 10^\circ$ is irrational. Any ideas?
0
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2answers
43 views

Fixing the closed form of $\sum_{k=1}^nk\sin^2(kx).$

I've been working on finding the closed form of this:$$\sum_{k=1}^nk\sin^2(kx).$$ Using the fact that:$$\sum_{k=1}^nku^k={u\over (1-u)^2}\bigg[nu^{n+1}-(n+1)u^n+1\bigg]\forall u\ge 1\quad (1)$$ I ...
5
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4answers
344 views

Trigonometric identities tan(x/2)

I have this task. I know that i) is $\displaystyle\frac{2t}{1-t^2}$ How do I get to ii) and iii) If $\displaystyle\tan(x) = \frac{2t}{1-t^2}$ I would multiply by $\cos(x)$ to get ...
2
votes
4answers
59 views

Extracting $x$ from $\cos(\arcsin(x))$

The following I know to be valid: $x = \sin(\arcsin(x))$ But is it possible to extract $u$ from $\cos(\arcsin(u))$ ? Should it be: $\cos(\arcsin(t)) = \sin\left(\dfrac{\pi}{2} + \arcsin(t)\right) ...
1
vote
3answers
31 views

Maximizing sin(a-b) given a trig relation

Suppose $a$, $b$ are acute angle measures such that $\tan a = 5\tan b$. Find the maximum value of $\sin(a-b)$. $\sin(a-b)=4\sin b \cos a$, but I don't know what to do from here.
1
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4answers
56 views

Evaluating $\lim_{x\to \infty} \frac{x - \sin(x)}{x+\sin(x)}$ [on hold]

How to find the value of $$\lim_{x\to \infty} \frac{x - \sin(x)}{x+\sin(x)}$$
0
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0answers
30 views

Calculating the length of a circular arc

In the post, How do the power-series definitions of sin and cos relate to their geometrical interpretations?, I am having trouble following the logic the blogger uses in the "Calculating the length of ...
-5
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0answers
40 views

What are the integration of these inverse trigonometric function? [on hold]

Integrate the following: Please Help me, I don't where to start. I used several methods to solve this like completing the squares.. $\int\frac{u^4+4}{u^4+9}du$ $\int\frac{\sin(x)(\cos ...
7
votes
2answers
252 views

Finding an inverse trigonometric sum

How do I prove that the following equality holds- $$\sum_{p=1}^{10} \sum_{q=1}^{10} \arctan \left(\dfrac{p}{q}\right)=25\pi$$ I tried to create telescoping terms by using the $\arctan{A}-\arctan{B}$ ...
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4answers
69 views

What is the maximum value of $f(\theta) = \sin\theta \cos\theta$

What is the maximum value of $f(\theta) = \sin\theta \cos\theta$ ?
0
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0answers
18 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
49 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 ...
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2answers
32 views

Integrate using trigonometric substitutions: [on hold]

Integrate $\frac {\sqrt{4x^2+4}}{x} $ using trigonometric substitutions
-6
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3answers
38 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
66 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
12 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
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2answers
24 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
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4answers
53 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 ...
4
votes
3answers
52 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}\left[\, {x^{3} \over \left(\,x^{2} + ...
4
votes
2answers
59 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
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1answer
40 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
41 views

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

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
59 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
106 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 ...
1
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1answer
51 views
+100

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
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3answers
49 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
55 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: ...
10
votes
5answers
199 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
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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
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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
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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 ...
9
votes
4answers
248 views

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

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
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
114 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!
1
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2answers
67 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 ...