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

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Simple expressions for $\sum_{k=0}^n\cos(k\theta)$ and $\sum_{k=1}^n\sin(k\theta)$? [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in A.P? I'm curious if there is a simple expression for $$ 1+\cos\theta+\cos 2\theta+\cdots+\cos n\theta ...
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1answer
488 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
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1answer
183 views

Prove $\frac{\cos^3{x}-\sin^3{x}}{\cos{x}-\sin{x}} =1+\frac{1}{2} \sin{2x}$

Prove $$\frac{\cos^3{x}-\sin^3{x}}{\cos{x}-\sin{x}} =1+\frac{1}{2} \sin{2x}$$ How do I start :( which identity do I use?
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1answer
59 views

Does the area of web chart diagrams depend on the order of items?

I am trying to discourage people from using radar chart diagrams (e.g. this one) where this is not appropriate. I have already made the point that the optical effect will be distorted because ...
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1answer
110 views

Finding a function that computes a point on the unit-circle

I can't find the definition of a function $f(x); x \in [-1;1]$ where $(x|f(x))$ is a point on the unit-circle. Can you please give me a hint? ---------- Edit ----------- Background: I want to ...
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478 views

Period of a product of $\sin$ and $\cos$

I want to find the period of $\sin(t) \cos(\pi t)$. I started off by transforming that into $\frac{1}{2}\left [ \sin((\pi +1)t) - \sin((\pi - 1)t\right ]$, but then I get stuck. How do I find the ...
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1answer
237 views

Prove that a transcendental function has exactly one root on a given interval

This is a follow up question from this thread Horizontal and vertical asymptotes of polar curve $r = \theta/(\pi - \theta) \, , \, \in[0,\pi]$ I need to show that the function $$ ...
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38 views

Is it okay to use the same variable to describe function periods?

If I have a system of period functions of $x$ and $y$, in this case trigonometric, $$\begin{cases} \sin{(2x + y)} = 0 \\ \sin{(2y + x)} = 0 \end{cases}$$ is it okay for me to to use the ...
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3answers
96 views

Evaluate a complex set

Can you please help me finding an exact description of the set: $$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$ For any $0<R \in \mathbb{R}$. My feeling is the $E_R = \mathbb{C}$, for any ...
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1answer
75 views

How to turn this sum to a product?

I arrived at this sum or trigonometric functions that I need to turn into a product in order to continue the exercise. How do I do that? $$\sin{x}\cos{(x+y)} -\cos{x}\sin{(x+y)}$$ I know of the ...
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3k views

Why is it that when proving trig identities, one must work both sides independently?

Suppose that you have to prove the trig identity: $$\frac{\sin\theta - \sin^3\theta}{\cos^2\theta}=\sin\theta$$ I have always been told that I should manipulate the left and right sides of the ...
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1answer
2k views

Horizontal and vertical asymptotes of polar curve $r = \theta/(\pi - \theta) \, , \, \in[0,\pi]$

I as supposed to find the vertical and horizontal asymptotes to the polar curve $$ r = \frac{\theta}{\pi - \theta} \quad \theta \in [0,\pi]$$ The usual method here is to multiply by $\cos$ and ...
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3answers
473 views

'Cosine'-esque function with flat peaks and valleys

I came up with this function: $$2\left(\frac{1}{1+e^{\textstyle\frac{-6\sin^{-1}(\cos(x))}{\pi/2}}}-\frac12\right)$$ to mimic a 'cosine'-esque function with flat peaks and valleys. Here it is as ...
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2answers
42 views

Simplifying this equation (trigonometric)

I was reading my notes and I came across these 2 lines, Im wondering how did it go from $\sin$ to $\sinh$? $-5\sin i\pi$ $ = -5i\sinh \pi$
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0answers
111 views

Set of all points which are a specified angle away from a given point on a sphere.

I have a sphere with a known point on the surface in polar coordinates. I'm looking to find the set of all points which are exactly some angle away from this point in polar form (this should describe ...
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2answers
93 views

Trigonometrical functions

I've been trying to solve this for 1hour, but I can't do it. I don't know if I'm missing something or it just doesn't open to me... Simplify the following expression: $$\frac{\sin^2 x - \cos^2 x}{ ...
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544 views

Which trigonometric identities involve trigonometric functions?

Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled functional equation gave the identity $$ \sin^2\theta+\cos^2\theta = 1 ...
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1answer
74 views

To what function's outputs do these numbers belong?

I have a sequence of numbers which have been used in an open source R Package. I am trying to understand how the package is accomplishing its objectives. So i need to know the function to whose set of ...
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1answer
211 views

The Cosine Rule

What is actually the Cosine rule. can anyone explain it to me in way that I can understand it, explain in a simple way? that provide simple examples? (Only the Cosine rule) thanks in advance, I tried ...
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1answer
120 views

are these angles considered equal, equivalent…?

We have an angle $\alpha$, then we have $\alpha +2k\pi$. How are these angles considered between themselves? When we apply trigonometric functions they have the same value but that is not always true ...
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2answers
110 views

prove$\frac{ \sin a\vphantom{(}}{\sin b} +\frac{\cos a\vphantom{(}}{\cos b} = \frac{2\sin (a+b)}{\sin 2b}$

I've got this far but don't understand where the $2$ on the numerator comes from: $$\dfrac{\sin a \cos b + \cos a \sin b}{\sin b \cos b}\overset{?}{=}\dfrac{\sin(a+b)}{\sin 2b}$$
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1answer
98 views

Solving trigonometry equation

I have a function in the form of: $$\cos^{-1} \left(\dfrac{a^2+ bx^2}{2abx}\right) + \cos^{-1} \left(\dfrac {c^2 + dx^2}{2cdx}\right) = e$$ How do I solve for $x$ if all other variables ($a$ through ...
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1answer
698 views

Finding the radius of a rectangle based on a given x,y coordinate.

Is it possible to find the radius of a rectangle based on a given x and y coordinate? If so, how is the achieved. I'm trying to create a rectangle in C++ ( using OpenGL ), and would like to obtain ...
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5answers
415 views

Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?

In proof of $\displaystyle\lim_{x\rightarrow0}\frac{\sin{x}}{x}=1$ is assumed that $\sin{x}\leq{x}\leq\tan{x}$ while $0<x<\frac{\pi}{2}$. First comparison is clear, arc length must be greater ...
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2answers
575 views

Evaluating $\int\sin^3t \, dt$

I have this integral: $$\int\sin^3t \, dt$$ I have tried partial integration with $\sin t \cdot \sin^2t$, but then I get another integral to evaluate which needs partial integration: $$\dots \int ...
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2answers
2k views

Using a Calculator with Decimal Degree Measures

So I have a problem with what I am being asked to do. Normally, when solving for the degree measures of trigonometric functions, I am presented with a fraction of rational numbers or a fraction with a ...
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3answers
144 views

Finding the Value of a Trigonometric Function

I am trying to solve a homework problem that has to do with deciding which of two trigonometric functions is greater. This would be simple to do with a calculator, but the instructions explicitly say ...
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3answers
109 views

Inequality for $\cot$

How can I prove that for all $t\in[0,\frac{\pi}{2}], \cot^2t\leq\frac{1}{t^2}\leq1+\cot^2t$, with $\cot$ the cotangent function ? Thank you
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1answer
114 views

The graph of $\cot$ is the image of the graph of $\tan$ by a simple transformation

How can I justify that the graph of the function cotangent : $\cot$ is the image of the graph of the function $\tan$ by a simple transformation.
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1answer
180 views

Rearrange $y = x\tan y$ to solve for $y$ given $x$

I have an equation: $$x = \frac{y}{\tan y}$$ or rewritten as: $$y = x\tan y$$ How can I rearrange this so that I can calculate $y$ given $x$?
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169 views

How to use a Vector Triangle to get values from U and V components.

I am trying to figure out how to get the heading of a wind with U and V components, using a Vector Triangle. I have been trying to study the bottom of http://www.aprweather.com/pages/wind.htm and ...
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2answers
144 views

Changing the argument for a higher order derivative

I start with the following: $$\frac{d^n}{dx^n} \left[(1-x^2)^{n+\alpha-1/2}\right]$$ Which is part of the Rodrigues definition of a Gegenbauer polynomial. Gegenbauer polynomials are also useful in ...
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8answers
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Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$?

$$\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$$ Is there any formula that tells this or why is it like that?
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1answer
261 views

How do we derive the direction formula for longitude latitutude

θ = atan2( sin(Δlong).cos(lat2), cos(lat1).sin(lat2) − sin(lat1).cos(lat2).cos(Δlong) ) Does it take into account that we may be dealing with a trapezoid rather than a rectangle ...
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1answer
313 views

Calculating $\sin(10^\circ)$ with a geometric method

Excuse me if this is a simple question: What is a simple geometric method for calculating $\sin(10^\circ)$ using only the sines of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$? Generally, is ...
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5answers
2k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
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1answer
75 views

Trigonometric term in digamma function $\psi_{0}(-n)$

Solutions to expressions s.a. $$ S(n)=\sum_{k=1}^{n}\frac{1}{k-r} = \psi_{0}(n-r+1)- \psi_{0}(1-r), $$ involves digamma function. For positive values it has the largest term $O(\log(n))$, but ...
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2answers
177 views

Evaluating $\int\sqrt{150^2-x^2} \cdot dx$

I'm studying for my finals and I have this integral that I'm trying to evaluate (part of a bigger problem): $$\int\sqrt{150^2-x^2} \cdot dx$$ I have evaluated a few integrals of this type before so ...
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2answers
208 views

Evaluate the definite integral: $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$

We were recently asked to evaluate this - $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$ I think we can start by breaking up the integral as $y(x) = \int_{0}^{\pi} \sin(x)\cos(y(x)) dx + \int_{0}^{\pi} ...
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1answer
1k views

distance of two point (latitude and longitude) on earth to meters without using $\cos,\sin, \mathrm{harversine}$ formula, etc.

I'm getting the distance between two locations (lat/long) using Pythagoras theorem. my data look like this (I use microdegrees because I have limitations) point1: -34608420,-58373160 point2: ...
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2answers
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trigonometry equilateral triangle

first of all, sorry for the lame question. Having a starting point, A and a height (catet) of y, what's the formula to calculate x? Thank you, i don't have any trig basis.
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135 views

is this equation solvable?

Can someone please solve these 2 equations to get values of h and k? I know the values of h and k but not sure how to solved these equations to get h and k 's values $(20.01 - h)^2 + (17.94 - k)^2 = ...
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4answers
2k views

How to get sine / cosine value out of tangens

I know that: $\tan(\alpha) = 1/2$. How can I get clean values for sine / cosine without the calculator? Is there a relationship? I know that $\sin(\arctan(1/2))$ is a way ... But I hope you get the ...
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3answers
489 views

finding point on line 1 unit next from start point

I have a line passing through points P1(x1,y1) and P2(x2,y2). Can I find next point on the same line thats 1 unit away from Point P1(x1,y1)? If yes how can I find? I just draw line at these points ...
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3answers
991 views

Newton's method and trig functions on a computer

I'm trying to use Newton's method to find roots for the function $A \cos(\Theta_2 - \Theta_1) + B \sin(\Theta_1)$. (That is, iterate $x_{i+1} = x_i - f(x_i) / f'(x_i)$). I've got a working ...
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1answer
2k views

how to find mid point of an arc?

I have start point $(x_1,y_1)$ and an end point $(x_2,y_2)$ and radius of arc. How to calculate the co-ordinates of mid-poing of arc? The arc is the part of a circle. Known Values ...
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2answers
267 views

Move an object along a straight path on an angle

I have an object at $x,y$ and I want it to move along a straight line on an angle of roughly $65^\circ$ and I know what the different of $X$ is but I do not know what the $Y$ should be. So for ...
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4answers
3k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
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3answers
108 views

calculating a point on circumference

See the diagram Known values are A: (-87.91, 41.98) B: (-104.67, 39.85) C: (-96.29, 40.92) L: 14.63 // L is OC Known angles ...
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4answers
469 views

Proving $a^2+b^2=c^2$ where $a,b,c$ are side lengths of a right triangle.

Proving $a^2+b^2=c^2$ where $a,b,c$ are side lengths of a right triangle. First, I have never done a proof before, sorry I am so poor here. I have spent many hours but my actions have mostly used ...