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

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2answers
152 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
2
votes
1answer
302 views

Singularities of $ \ \frac{z-1}{z^2 \sin z} \ $

Find all singularities of $ \ \frac{z-1}{z^2 \sin z} \ $ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
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1answer
142 views

Math Terminology: “Find all solutions” = “Solve the equation on interval 0 < x < 2π”

Suppose the following question: "Solve the equation sin(3x) = -1/2 on the interval 0 < x < 2π" Would you interpret the question as meaning: "Simply Solve the equation on the given ...
2
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1answer
608 views

Kinect skeleton scaling in 3d space

I am developing a physioterapy system with kinect and need to scale a skeleton size to another skeleton size. The kinect sensor recognizes 20 body joints, of every joint i have the x, y, and z ...
3
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5answers
250 views

Proving $|\sin x| \leqslant |x |$

Is it possible to prove $|\sin x| \leqslant |x |$ with only trigonometric identities? (not with well-known calculus or geometry proofs)
4
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1answer
89 views

Showing that $ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$

I'm trying to show that: $$ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$$ It is equivalent to show that: $$ (1-\cos x) \left (\frac{\sin ...
7
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1answer
262 views

Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$

I would like to show that: $$ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$$ We have: $$ \sum_{n=1}^N \arctan \left( \frac{2}{n^2} \right) =\sum_{n=1}^N \arctan ...
3
votes
2answers
223 views

Trigonometric equation

I have been trying to solve this equation for over a week now: $$\tan5x-2\tan3x=\tan3x\tan5x$$ I found one solution $x=k\pi$ but I cannot prove that this is the only solution. It is equivalent to: ...
1
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1answer
153 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
0
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1answer
47 views

How to solve the following special inequality?

Find $k$, as a function of $d_2$ and $d_3$, such that: $$\left \vert { d_2 \left [ \sin(e^{d_3\,y}) - \sin(e^{d_3\,x})\right] + (x-y) d_2 d_3 e^{d_3\,z} \cos(e^{d_3\,z})} \right \vert \le k ...
2
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2answers
136 views

Self-intersection of vector valued function

A vector valued function $r(t)$ has the following coordinates: $$x = 4\cos\left(\frac12t\right)+2\cos(2t)+\cos(4t)\\ y = 4\sin\left(\frac12t\right)+2\sin(2t)+\sin(4t)$$ I have to find the $t$-values ...
3
votes
1answer
299 views

Tricky integration by substitution $\int_{-1}^{1} \frac{ \sqrt{1-x^2}}{1+x^{2}} dx$

I have to get this integral $$\int_{-1}^{1} \frac{ \sqrt{1-x^2}}{1+x^{2}} dx$$ into $$\int_{-\pi }^{\pi } \frac{1}{1+\cos^2\theta } \,d\theta - \pi$$ any tips would be recommended.
2
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2answers
686 views

Prove $\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \csc^2 (2x)$

I'm having massive issues proving this identity: $$\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \csc^2 (2x)$$ How is this proven?
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2answers
63 views

Can you help me solve this ODE?

I need to solve this differential equation. YWhat I'm looking for is a way to simplify this equation. Can anybody give me hints/tricks to understand the following equation better: ...
0
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1answer
58 views

Perplexities on the Weierstrass substituition for $\phi=\pi$

Let us suppose that we have a system of equations including trigonometric expressions in $\phi$ and we want to bound the number of possible solutions. If I apply the Weierstrass substituition ...
1
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1answer
226 views

$\cos^n x\sin^m x$ as sum of sine and cosine functions of multiples of the argument.

Gradshteyn and Ryzhik (2007), e.q. 1.320 has some formulas for representing the powers of sine and cosine functions as sums of sine or cosine functions of multiples of the angle, e.g. $$\cos^3x = ...
4
votes
2answers
232 views

Finding a limit without series expansion and l'Hopital's rule

I have to find $$\lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3x}$$ without series expansion nor l'Hopital's rule, and am utterly and completely lost. I ended up putting $x = 2y$ and getting to ...
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3answers
174 views

Sum of two sine curves

How can we compute the sum $$ \sin(f_1) + \sin(f_2) $$ I know it is $$ 2\sin\left(\frac{f_2 + f_1}{2}\right) \cos\left(\frac{f_2 - f_1}{2}\right) $$ but how can it be derived with elementary ...
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1answer
2k views

General Cartesian/Rectangular Equation for Polar Rose ($r=\sin(k\theta)$)

How do I convert the Polar Equation $r=\sin(k \theta)$ to Cartesian Equation? I understand that $r^2=x^2+y^2$ and that $x=r\cos\theta$ and $y=r\sin\theta$, but no matter how I try to arrange them it ...
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0answers
92 views

$\cot(x)$ or $\tan(x)$ amplitude with $F(x)$ or $G(x)$?

If you are doing $f(x)$ and $g(x)$ of a tangent/cotangent function and you get an amplitude. Should you write the final equation with or without the amplitude because technically tan and cot don't ...
1
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1answer
180 views

How to simplify geodetic distance formula?

Question is also related to programming, but maybe I can solve it here. I am using distance calculation using Haversine method(probably it's not important, but I will post the function): ...
1
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1answer
129 views

pre calc, sinusoidal equations

If the sea level decreases leaving the seabed exposed (normally $30$ feet below sea level), then it rises a equal distance above sea level. waves have a maximum height of $38.9$ meters. the cycle of ...
2
votes
2answers
71 views

simple question involving trigonometry

Can anybody explain what $$\tan(\sin^{-1}(\frac x y))$$ equals? I have to determine whether $$y'' \left(\tan \left(\sin^{-1}\left(\frac x y \right)\right) - \frac{x}{\sqrt{y^2-x^2}} \right)=0$$ is a ...
1
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3answers
2k views

Integral of cosec squared ($\operatorname{cosec}^2x$, $\csc^2x$)

According to my sheet of standard integrals, $\int \csc^2x \, dx = -\cot x + C$. I am interested in a proof for the integral of $\operatorname{cosec}^2x$ that does not require differentiating $\cot ...
0
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2answers
233 views

Chasing Problem [duplicate]

Possible Duplicate: Four turtles/bugs puzzle Starting from the corners of a square of side a, each bug chases the one clockwise from it. If they all start at the same time and run at the ...
1
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1answer
352 views

Cauchy product and the exponential function

Simplify the following series using the Cauchy product $$\sum\limits_{k=1}^\infty\frac{1}{k!}\cdot\sum\limits_{j=1}^\infty\frac{1}{j!}$$ ...
2
votes
2answers
385 views

Trigonometric general solution to ordinary differential equation

Solve: $$\frac{dx}{dy}=(x^{2}-x-12)(1+\tan^{2}{y})$$ This is a first order, linear, separable ODE, so it can be solved by rearranging to: $$\frac{dx}{x^{2}-x-12}=(1+\tan^{2}{y})\:dy$$ And then ...
3
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1answer
846 views

Solution Sets of Trigonometric Equations

Introduction Hi there. In advance, I apologize if this question is too broad. Please do not downvote if that is the case, as this question is purely imaginative curiosity. I will close it should it ...
2
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2answers
2k views

How do you find the equation for the angle bisecting line given three coordinates that make up an angle?

I have three points,$$A =[A_x,A_y]\,,\, B =[B_x,B_y]\,,\,C =[C_x,C_y]$$ How could one calculate the equation for the line that bisects the angle $\,\angle ABC\,$ (eg., passing through $B$)?
8
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3answers
207 views

Other ways of solving $\cot^{-1}(x)=\sin^{-1}(x)$

Real solutions to $$\cot^{-1}(x)=\sin^{-1}(x)$$ I found this problem in an exam years ago and I solved it using geometry. The first mistake I made was assuming ...
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2answers
24 views

Trigonometric development help.

I need help with the following trigonometric development: $ x = r(\theta)\cos\theta$ $ y = r(\theta)\sin\theta$ this gives: $ x' = r'(\theta)\cos\theta - r(\theta)\sin\theta$ $ y' = ...
3
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3answers
239 views

Are trig identities commutative?

If I have $\cos(B)\cos(A)-\sin(A)\sin(B)$, can I write that as $\cos(A)\cos(B)-\sin(A)\sin(B)$? And then combine it as $\cos(A+B)$?
1
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4answers
491 views

Geometry Prove - two perpendicular lines in a circle

In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X. Show that:
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1answer
29 views

Finding The Obtuse Angle

I asked a question on physics stackexchange, that I suppose would be more appropriate for the math forum: http://physics.stackexchange.com/questions/44229/finding-the-obtuse-angle I'd appreciate your ...
1
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2answers
120 views

area of a circle - 3/4th

How to find the pixels of that line which is crossing the circle? Is there any formula? Iam getting the line's end points
4
votes
2answers
102 views

Geometric identity, cannot show equivalence using trigonometric identities

clearly $$(x+a \cos\theta)^2+(y-a \sin\theta)^2=b^2$$ expanding and using the Weierstrass substitution we find that $$\theta= 2 \arctan \frac{\left( 2ay- \sqrt{ 4a^2y^2 - ( (x-a)^2+y^2-b^2)( ...
2
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2answers
298 views

Help With Trig Compound Angles: identity for $\cos(\theta - 60^\circ)$?

$$\cos(\theta-60^\circ)=\frac{1}{2}\sin(\theta)$$ $$2\cos(\theta - 60^\circ)=\sin(\theta)$$ $$2=\frac{\sin(\theta)}{\cos(\theta-60^\circ)}$$ How do I get rid of the $60^\circ$ so that I can make this ...
1
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1answer
301 views

Trig identity manipulation question

I'm working on manipulating trig identities and using Wolfram Alpha to check the identity still holds. I'm going from this: $$\frac{1-\cos x}{1+\cos x} = \frac{1}{tan^2x}-\frac{2}{\tan x \sin x} + ...
1
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3answers
384 views

Help With Double Angles And Trig Identity Problem

Hello please help me with these trig identities and double angles as I am not sure where I am going wrong but I keep getting the wrong answer This is the problem $$ \sin(\theta+30) = 2\cos(\theta) ...
2
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3answers
154 views

How do I rearrange this formula? Circles around a larger circle.

My A-Level algebra is failing me. Can someone please tell me how to rearrange this formula to give $n$ when you know $R$ and $r$. $R \sin(180^\circ/n)/(1 - \sin(180^\circ/n)) = r$ This formula is ...
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1answer
146 views

General solutions of Trigonometric problem.

There are some literature to say the solutions for all trigonometric ratios. But the following problem is given in particular interval. Kindly answer this question. In the interval $x$ in $[0, ...
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0answers
60 views

Finding the smallest integer n [duplicate]

Possible Duplicate: Proving that $ ...
0
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1answer
98 views

Solving for $x$ in $x(t) = \frac{-2}{3}\cos(10t) + \frac{1}{2}\sin(10t)$

A physics problem is asking me a to find when a weight on a spring crosses the equilibrium point. The equation of motion given is $$x(t) = \frac{-2}{3}\cos(10t) + \frac{1}{2}\sin(10t)$$ Basically, ...
0
votes
1answer
358 views

Pre calc sinusoidal function word problem

A car travelling 18km/hr drives over a nail and it sticks in one of the front tires. The tire has a radius of 14 cm. Determine the height of the nail above ground 5 minutes after the car drives over ...
4
votes
1answer
130 views

Regular polygons and Pythagoras

Let $L_n:$ the side length of a regular $n$-polygon inscribed in a unit fixed circle. We have an interesting relationship: $L_6^2+L_6^2=L_4^2$ $L_6^2+L_4^2=L_3^2$ $L_{10}^2+L_6^2=L_5^2$ There ...
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3answers
103 views

Solution of trigonometric equation $at + \sin(t) = 0$

Let $a \in \mathbb R$, what values of $t$ solve the equation $at + \sin(t) = 0$?
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1answer
202 views

Figure out position with compass bearing

Scenario: You have two people in the same room, both transmitting their compass bearing. Question: Is it possible to figure out which direction the other person is standing just from the compass ...
0
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0answers
131 views

Calculate distances with Lat and Long

I am trying to calculate a waypoint a set distance away from my current location. I know my Lat and Long of my current location and the distance away a want the waypoint to be. I also know the ...
1
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2answers
624 views

A cubed trigonometric identity?

Could somebody please show why the following is a trigonometric identity? $$\dfrac{\sin^3 a - \cos^3a}{\sin a - \cos a} = 1 + \sin a \cos a$$ This problem appears on page $48$ of Gelfand's and ...
0
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1answer
112 views

Finding a diagonal of a trapezoid that touches 3 points on a circle

In the image below: - AB and AD are tangent to the circle - BC and AD are parallel What is the length of AC? Thank you very much in advance!