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

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13
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3answers
930 views

Expressing $\sin(2x)$ as a polynomial of $\sin{x}$

Using trignometric identities ( double angle forumlas) one can see that $\cos{2x} = 2 \: \cos^{2}{x} - 1$ can be expressed as a polynomial of $\cos{x}$, where $p(\cos{x})=2 \: \cos^{2}{x}-1$. Then its ...
1
vote
1answer
509 views

Relation between “harmonic form” and fourier series?

I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare ...
14
votes
4answers
3k views

Aunt and Uncle's fuel oil tank dip stick problem

This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I ...
7
votes
1answer
2k views

A series expansion of $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
6
votes
4answers
719 views

What are some good resources for brushing up on geometry and trigonometry?

I'm brushing up on my calculus, and geometry and trigonometry keep coming up. However, these are probably my weakest areas mathematically. Are there any good sites or free PDFs/ebooks that I can use ...
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...
2
votes
7answers
1k views

How many ways to define sine and cosine?

Sometimes, there are many ways to give defintion to a mathmatical principle, for example the natural base logarithm. How about sine and cosine? thanks.
4
votes
3answers
3k views

Find the coordinates in an isosceles triangle

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
2
votes
2answers
375 views

Approximating a cosine

Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$. Given that definition, if I introduce a new variable $t$ is the following a ...
4
votes
1answer
673 views

Finding equal sums of constrained trig functions

Problem 63 of the 2001 St. Petersburg Mathematical Olympiad, Second Round, 11th grade: Are there three different numbers $x, y, z$ in $[0,\pi/2]$ such that the numbers $\sin x$, $\sin y$, $\sin z$, ...
3
votes
3answers
9k views

Rotate a point in circle about an angle

How should I rotate a point $(x,y)$ to location $(a,b)$ on the co-ordinate by any angle?
19
votes
4answers
1k views

Prove that $\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$

How can you prove that: $$\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$$ for every integer $n\geq 1$. PS: no, it's not a homework... :-)
13
votes
1answer
464 views

Is $\sin(n^k) ≠ (\sin n)^k$ in general?

Is it true that $\sin(n^k) ≠ (\sin n)^k$ for any positive integers $n$ and integers $k ≠ 1$? What if $n > 0, k ≠ 1$ are rational?
3
votes
4answers
8k views

How to calculate reflected light angle?

On a 2D plane, line X is at x radians angle, an incoming light travels at y radians angle, how to calculate the angle of the outgoing light reflected off line X? How to do this in a way to cover all ...
5
votes
2answers
382 views

Rigorous synthetic geometry without Hilbert axiomatics

Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other ...
32
votes
7answers
10k views

How can I understand and prove the “sum and difference formulas” in trigonometry? (cos(a ± b) = …, etc.)?

The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true. \begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha ...
7
votes
7answers
1k views

Different definitions of trigonometric functions

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two ...
5
votes
3answers
1k views

Law of cosines with impossible triangles

Is there any mathematical significance to the fact that the law of cosines... $$ \cos(\textrm{angle between }a\textrm{ and }b) = \frac{a^2 + b^2 - c^2}{2ab} $$ ... for an impossible triangle yields ...
6
votes
1answer
2k views

RHS Congruency test - What makes 90 degrees different?

RHS is a well known test for determining the congruency of triangles. It is easy enough to prove it works, simply use Pythagorus' theorem to reduce to SSS. I thought that it seems strange that this ...
4
votes
3answers
1k views

How do I measure distance on a globe?

I have a $3$-D sphere of radius $R$, centered at the origin. $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are two points on the sphere. The Euclidean distance is easy to calculate, but what if I were to ...
6
votes
1answer
896 views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
10
votes
2answers
2k views

What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
25
votes
15answers
5k views

Intuitive understanding of the derivatives of $\sin x$ and $\cos x$

One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) ...
15
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10answers
13k views

Real world uses of hyperbolic trigonometric functions

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses ...