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

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2
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2answers
153 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 ...
6
votes
8answers
3k views

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?
0
votes
1answer
269 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 ...
9
votes
1answer
315 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 ...
5
votes
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 ...
0
votes
1answer
76 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 ...
3
votes
2answers
178 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 ...
2
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2answers
211 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} ...
1
vote
1answer
2k 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
3k views

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.
0
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2answers
136 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 = ...
1
vote
4answers
3k 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 ...
1
vote
3answers
559 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 ...
2
votes
3answers
1k 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 ...
3
votes
1answer
3k 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 ...
0
votes
2answers
310 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 ...
13
votes
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 ...
1
vote
3answers
113 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 ...
2
votes
4answers
529 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 ...
2
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5answers
2k views

finding center of circle

How can I calculate center of a circle $x,y$? I have 2 points on the circumference of the circle and the angle between them. The 2 points on the circle are $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The ...
1
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3answers
2k views

how to draw an arc?

How to draw an arc, I have these values ...
1
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1answer
296 views

What is the process in calculating values on the Unit Circle?

I realize that this question is incredibly basic for this website, but I really need help. I took this image from MathIsFun.com: It's a picture of the Unit Circle. On the outside, in purple, are ...
0
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1answer
180 views

How to calculate the angle at which a ship moves?

Knowing the angle of the sail and the angle of the wind, how can you calculate the resulting angle of the boats movement?
4
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1answer
161 views

Trigonometric expressions possible pattern

While solving trigonometric problems I've noticed possible pattern that reminded me of Fibonacci numbers and Pascal triangle. So I tried to find next "element" of this pattern (8 degree exponent ) and ...
10
votes
1answer
550 views

Proving that $ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+…+\frac{1}{\sin(133°)\sin(134°)}=\frac{1}{\sin(1°)}$

I would like to show that the following trigonometric sum $$ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+\cdots+\frac{1}{\sin(133°)\sin(134°)}$$ ...
1
vote
1answer
212 views

Ancient astronomers, planetary conjunctions, and epicycles

How did ancient astronomers predict planetary conjunctions? I know they used a system of epicycles to represent the path of planets, but finding the point and time of alignment of two planets still ...
-1
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1answer
131 views

Find new Lat/Long if Lat/Long and constant distance is given

I have a list of coordinates of latitude and longitude which make an 'S' shape if I join the coordinates sequentially. My task is to make a parallel 'S' translated 0.1 nautical miles from the original ...
1
vote
1answer
309 views

Sum of sinusoidals (frequency/phase of acoustic beats)

I have a function that's the sum of two sinusoidals: $ A \cos(\Theta_1 + \omega_1 t) + B \cos(\Theta_2 + \omega_2 t) $. It basically forms an acoustic beat pattern. I need to find the frequency of ...
1
vote
2answers
129 views

Computing $ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x°)}\right) $

I would like to calculate: $$ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x°)}\right) $$ Here is what I found: $$ 1-\frac{1}{\tan(x)}=\frac{\sqrt{2}\sin(x-45°)}{\sin(x)} $$ $$ ...
38
votes
6answers
3k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
3
votes
6answers
3k views

calculating angle in circle

How to calculate angle in a circle. Please see the diagram to get the idea what I want to calculate? I have origin of circle that is $(x_1,x_2)$. I have a point on circumstance of circle that is ...
6
votes
3answers
415 views

Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute

As the question indicates we are supposed to find the modulus of z-1. When trying to solve the problem I drew a diagram which you can see below: The book I am working in solved a similar problem ...
3
votes
4answers
597 views

Simple trigonometry question (angles)

I am starting again with trigonometry just for fun and remember the old days. I was not bad at maths, but however I remember nothing about trigonometry... And I'm missing something in this simple ...
1
vote
2answers
139 views

Calculate an angle from Tan, Cos, or Sine

I'd like to calculate an angle using nothing but either tangent, cosine, or sine, or any combination of the two. Is this possible? If so, how? Why I'm asking this I know how to program, but I find ...
4
votes
2answers
381 views

Wrong initial intuition about the limit of $\frac{\sin(x)}{x}$ as $x$ goes to 0

I'm following the MIT OpenCourseware course on calculus, and when proving the derivative of $\sin(x)$ the following assumption is needed $\lim\limits_{x\to0} \frac{\sin(x)}{x} = 1$ The proof for ...
4
votes
1answer
1k views

Finding force for trajectory

Using the trajectory equations provided on hyperphysics, I have developed some code to plot the trajectory of an object. I now need to work out the x and y forces to apply to make the object follow ...
3
votes
1answer
530 views

Functional inverse of $\sin\theta\sqrt{\tan\theta}$

What is the functional inverse of $f(\theta) = \sin\theta\sqrt{\tan\theta}$? Or, equivalently, what is the inverse of $$f(\theta)=\sin^2\,\theta\tan\,\theta=\frac{\sin^3\,\theta}{\cos\,\theta}$$ It ...
3
votes
1answer
140 views

Graphically display $a^3+b^3+c^3 = d^3$

As a kind of 'addition' to Fermats Last Theorem, a friend of mine has come up with a different idea. Let $$a^3+b^3+c^3 = d^3$$ with $(a,b,c,d): a,b,c,d \in \mathbb{N}$. We were discussing Pythagoras: ...
11
votes
4answers
8k views

How to remember the trigonometric identities

I have a test tomorrow and I am having trouble remembering those pesky trigonometrical identities (such as $1-\cos x=2\sin^2(\frac{x}{2})$ ) Do you guys have any tips on how I can remember these? ...
4
votes
4answers
399 views

Why is $\lim\limits_{x \space \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$?

As part of this problem, after substitution I need to calculate the new limits. However, I do not understand why this is so: $$\lim_{x \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$$ I tried ...
7
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1answer
277 views

Solving $\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$

I'm trying to solve $$\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$$ By substituting $x=2\tan{t}$. I get as far as: $$\int_{x \space = -\infty}^{x \space = ...
1
vote
0answers
5k views

A complicated trigonometric equation

I have the following trigonometric equation $$f(\theta)=100(A_2 B_3 - A_3 B_2)^2 - (c_1B_3 - c_2 B_2)^2 - (c_2A_2 - c_1 A_3)^2=0,$$ where: $ A_2 = 3\cos(\theta)-5$ $B_2 = 3\sin(\theta)$ $A_3 = ...
3
votes
1answer
660 views

Proving formulas for $\cos(nx)$ and $\sin(nx)$

How do I prove the following formulas? Let $n \in \mathbb{N}, x \in \mathbb{R}$. Prove that: $$\cos(nx)=\sum_{j=0}^{[n/2]} (-1)^j {n \choose 2j} (\cos x)^{n-2j} (\sin x)^{2j}$$ ...
10
votes
6answers
3k views

Relationship between $\tanh x$ and $\arctan x$

The functions $\tanh x$ and $\arctan x$ have a similar graph. Is there a formula to transform $\tanh x$ to $\arctan x$?
3
votes
2answers
245 views

Losing information when solving trig problem

I was doing a simple trig question when it turned out I was missing several answers. I have read somewhere that it is possible to lose information about the signs when dealing with squares and square ...
0
votes
4answers
150 views

How do I find the solutions of this equation?

How do I find the solutions of this equation: $$\tan^2 (x)=-1$$
2
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7answers
265 views

Unconventional way to solve trig question

I was working on a trig question and got stuck, but then I noticed a possible way to solve the problem. However, this way seemed to be slightly unconventional and possibly not what the book was ...
2
votes
6answers
602 views

Intuitive Explanation of the graph $y = \sin x$ [duplicate]

Possible Duplicate: Intuition for graphing Sine/Cosine We've all seen the graph of $y = \sin x$ (I can't post an image because of reputation so I posted a link to a graph) Sine Graph ...
3
votes
2answers
369 views

Simplifying trig expression

I was working through some trig exercises when I stumbled upon the following problem: Prove that: $ \cos(A+B) \cdot \cos(A-B)=\cos^2A- \sin^2B$. I started out by expanding it such that $$ \cos(A+B) ...
0
votes
2answers
192 views

Solving trig equation with $\sin A$ and $\sin 2A$

I am stuck trying to solve for $A$ in $$3 = 11\sin^2 A - 2\sin 2A$$ I cannot see a way to manipulate to get like terms and hence factor it. Thanks!