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

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5
votes
3answers
168 views

Limit of $( n\sin^2(x\pi)-n\sin^2(\sqrt{n}\pi))/(x-\sqrt{n})$ without using L'Hopital

I was asked to prove this , without using L'Hopital... tried out some trig. identities with no big use $(\sin(\alpha)-\sin(\beta))(\sin(\alpha)+\sin(\beta))=\sin^2(\alpha)-\sin^2(\beta)$ for example, ...
6
votes
2answers
295 views

How to prove that $\int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$?

I am stuck trying to show that $\displaystyle \int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$ I have tried using a Squeeze Theorem type approach, but at $\pi/4$ any function I choose overlaps ...
3
votes
3answers
2k views

Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$ [duplicate]

Possible Duplicate: Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? How to prove $$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$$
3
votes
4answers
39k views

Given the base and angles of an isosceles triangle, how to find length of the two sides?

I can't seem to find a textbook solution to this. It is always assumed that the length of the sides is know. Isolceles triangle So the base $a$ is known. The bottom angles where $\alpha$ and the ...
2
votes
1answer
172 views

Method for finding roots of real trigonmetric polynomial

Given a real valued trigonometric polynomial, $$ f(x) = \sum_{k=0}^{n} a_k \cos(k x + \phi_k) $$ what is the current fastest numerical method to find the roots of the polynomial for a given error? I ...
1
vote
0answers
72 views

Help with manipulating a change of variable in Integration

Knowing: $$\phi (x)=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ \sqrt { 1-{ x }^{ 2 } \sin ^{ 2 }(t) } } } $$ I am trying to demonstrate that: $\phi (x)=\frac { 1 }{ 1+x } \phi \left( ...
4
votes
3answers
160 views

Help in manipulating Integrals

I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $ as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $ I tried to use the exponential function ...
2
votes
4answers
79 views

Calculate value of expresion E(x)

Calculate the value of expresion: $$ E(x)=\frac{\sin^6 x+\cos^6 x}{\sin^4 x+\cos^4 x} $$ for $\tan(x) = 2$. Here is the resolvation but I don't know why $\sin^6 x + \cos^6 x = ( \cos^6 x(\tan^6 x + ...
4
votes
3answers
574 views

how do you learn trigonometric identities [duplicate]

Possible Duplicate: Is there a more efficient method of trig mastery than rote memorization? i find myself loosing it in 1st semester calculus, mainly because people are using trigonometric ...
3
votes
2answers
170 views

The integral $\int\frac{1+\sin x}{\cos x}dx$

Is $$\int\frac{1+\sin x}{\cos x}dx$$ the same as the integral of $$\sec x+\tan x$$ (since $1/\cos x = \sec x$ and $\sin x/\cos x = \tan x$)?
3
votes
4answers
7k views

Integral of $\sin(\sqrt{x})$

I need help finding the integral of $\sin(\sqrt{x})dx$. I have the answer here but would like to know how to get there.
3
votes
1answer
82 views

Check If a point on a circle is left or right of a point

What is the best way to determine if a point on a circle is to the left or to the right of another point on that same circle?
1
vote
3answers
77 views

How can i show this equality in trigonometry?

Help me to prove that $$\arcsin(\sqrt{2}\sin(t))+\arcsin(\sqrt{\cos(2t)})=\pi/2$$ Starting here is a bit difficult. Thank you in advance.
1
vote
2answers
363 views

How to get coordinates of point knowing distance from x,y and angle?

I have such a problem : I am given : x,y $\|a\|$ $\alpha$ $\vec{v}$ and $\|v\|$ I need to get the coordinates of point X1Y2.
-4
votes
2answers
148 views

As shown in the figure: Prove that $X=30.$

Any idea about this problem: As shown in the figure: Prove that $X=30.$
0
votes
2answers
66 views

Trigonometric relations

Determine the angle $v$ between $\pi/2$ and $\pi$ that meet $\cos v = \cos(23\pi/18)$. The answer should be able to be written like $v=a\pi/b$ where $a/b$ is a abbreviated fraction.
7
votes
1answer
393 views

Pretty solution to the trigonometric equation

Problem Consider the trigonometric equation: $$ a\sin x+b\cos x-\cos x\sin x=0\qquad(0\le x<2\pi)\tag{*} $$ try to analyze the number of solutions to equation (*) with parameters $a,b$, i.e, let ...
1
vote
2answers
292 views

Proving $x+\sin\sqrt{x}$ is uniformly continuous - not sure of my solution

Prove $x+\sin\sqrt{x}$ is uniformly continuous on $[0,\infty)$. Here's is my proof, but I'm worried of me $\delta$ choice being incorrect. If $x_1,x_2>0$ then ...
10
votes
3answers
742 views

$\tan(x) = x$. Find the values of $x$

How can I find the possible values of $x$ for: $\tan(x)=x$ mathematically?
3
votes
1answer
768 views

Distance from point to line using $x \sin \theta - y \cos\theta$

I am struggling to understand an equation given in an academic paper (in atmospheric sciences/geography) that I am reading. The paper defines a line, called the Clear Line, which is derived through ...
3
votes
2answers
343 views

Simple Trig Problem

I'm a bit stuck on a homework question that I've been assigned. The question is as follows: You are paddling a canoe at a speed of $4$ $km/h$ directly across a river that flows at $3$ $km/h$. ...
1
vote
2answers
75 views

Integration by substitution trig

I need to integrate $\frac {1}{2-\cos x}$ and I am given $t=\tan(x/2)$. What should I do with it?
2
votes
2answers
1k views

How do I find, by the definition of a derivative, the derivative of $\tan x$?

How do I find, by the definition of a derivative, the derivative of tanx? $$f'(x)=\lim_{\Delta x \to 0}{f(x+\Delta x)-f(x)\over \Delta x}=\lim_{\Delta x \to 0}{\tan(x+\Delta x)-\tan(x)\over \Delta ...
2
votes
4answers
662 views

How to derive inverse hyperbolic trigonometric functions

$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of ...
0
votes
3answers
1k views

Multiplication using Tangents.

I got a maths problem and just checking whether my answer and method is correct: Tan(A+B/2) / Tan (A-B/2) = SinA + SinB / SinA - SinB I started to solve it: ...
3
votes
1answer
2k views

Finding the sum of a trigonometric series [duplicate]

Find the sum of the series $$\cos x + \cos 2x + \cdots + \cos (n-1)x.$$ You must calculate the sum of this series only by multiplying through by $2\sin\left(\frac{x}{2}\right)$. Now I've heard of ...
2
votes
3answers
191 views

Isosceles triangle

Let $ \triangle ABC $ be an $C$-isosceles and $ P\in (AB) $ be a point so that $ m\left(\widehat{PCB}\right)=\phi $. Express $AP$ in terms of $C$, $c$ and $\tan\phi$. Edited problem statement(same as ...
3
votes
3answers
17k views

X and Y coordinates of circle giving a center, radius and angle

I have to find the necessary translations in X and Y to move a point 0n a circle to another one. I have a center (X and Y coordinates), a radius, and a current position in radians. And given a value ...
3
votes
1answer
2k views

Explanation of this image warping (bulge filter) algorithm

I've been researching image warping algorithms lately and haven't found many comprehensive references. That said, there are of course code snippets from GIMP, jhlabs.com, and imagemagick.org but none ...
1
vote
1answer
41 views

Compute angle from vertical at which a sphere strikes the lip of a cup

I'm working on a problem, wherein a sphere of known radius is dropped vertically and strikes the edge of a cup. I need to figure out the angle of deflection, which will be a function of where along ...
2
votes
2answers
1k views

Is the tangent function (like in trig) and tangent lines the same?

So, a 45 degree angle in the unit circle has a tan value of 1. Does that mean the slope of a tangent line from that point is also 1? Or is something different entirely?
3
votes
1answer
115 views

Trig question, can this be solved?

I was wondering is it possible to solve this without assuming that CAD=DAB. As I use the law of sines, trigonometry and have tried to apply law of cosines. However, I cannot see how you can solve ...
5
votes
4answers
645 views

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
1
vote
1answer
221 views

Find the acute angle $x$ for $\tan x = \tan(x+10^\circ)\tan(x+20^\circ)\tan(x+30^\circ)$.

How to solve the following equation? $$\tan x= \tan(x+10^\circ)\tan(x+20^\circ)\tan(x+30^\circ)$$
11
votes
3answers
307 views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is ...
6
votes
2answers
370 views

Help me solve this olympiad challenge?

Given: $$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$ What is the sum of all arctan of all the roots of $p(x)$?
3
votes
1answer
1k views

How to find the 3D coordinates on a celestial sphere's surface?

With celestial I don't mean a normal sphere, but I mean one that uses the altitude and an azimuth angle system. This is what I mean for example: (the star in the image represents an example of a ...
1
vote
0answers
58 views

Simple equation misunderstanding

Im trying to use an equation on this page http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics The angle of lean, $\theta$, can easily be calculated using the laws of circular motion $$ ...
16
votes
4answers
482 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
1
vote
1answer
33 views

Will this make sure a point does not lie along a line?

If I have $2$ vector points and I wish to create a 3rd vector point to make a plane from. I'd like to make sure the 3rd point I generate doesn't happen to have the same slope. I can add either $1$ ...
1
vote
1answer
206 views

Use the de Moivre theorem to evaluate the following

$$\frac{(1+i)(\sqrt{3} + i)^3}{(1-\sqrt{3}i)^{3}} = 1-i$$ What confuses me is how would I do the numerator because I have two expressions.
5
votes
4answers
237 views

Help Solving Trigonometry Equation

I am having difficulties solving the following equation: $$4\cos^2x=5-4\sin x$$ Hints on how to solve this equation would be helpful.
3
votes
2answers
734 views

Solving Trigonometry Polynomial Equation.

I am trying to understand how to solve the equation $$2\sin^2x + 3\sin x + 1 = 0.$$ Please give hints.
0
votes
1answer
97 views

Using polar form find product

How would I use polar form to show $$(-1-i\sqrt{3})(-4\sqrt{3}+4i)=8\sqrt{3}+8i$$ I tried putting it in polar form. And I got $$2(\cos(225)+i\sin(225))(2\sqrt{7}\cos(150)+i\sin(150))$$ But I keep ...
4
votes
1answer
134 views

On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
0
votes
2answers
277 views

What does negative sine mean in this diagram?

I thought cos was x and sin was y. In quadrant two, cos is negative and sin is positive. Why does this diagram have a negative sign as the x-coord and cos as the y coordinate for q prime's vector? ...
1
vote
2answers
177 views

Limit of a trigonometric function

I'm trying to find the following limit: $$ \lim_ {x \to 1} \frac{\sin{\pi x}}{1 - x^2} $$ I can't figure it out how to reach the fundamental trigonometric limit. Everything i see is that the ...
3
votes
1answer
70 views

What guarantees that non-geometric definition of trigonometry is actually the same as the geometric definition?

There are many different approaches to trigonometric functions, and it's clear that they share many properties. However, i don't know what guarantees nongeometric definition and geometric definition ...
4
votes
2answers
667 views

Find if a point is in a circle

I am coding a video game, but I am not so good at the math. I am hoping for some help here: Given: $X, Y$ that is the center of the Circle $R$ that is the radius of the Circle $X_1, Y_1$ that may ...
3
votes
1answer
4k views

Intersection between a rectangle and a circle?

I have a poor working knowledge of math. I would like to calculate collision detection between a 2D circle and a 2D rectangle for a simple game of Pong. I thought of splitting the 2D rectangle into 4 ...