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

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1answer
95 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
2
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1answer
91 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
2
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4answers
93 views

Solve $\tan\left(\arccos\left(\frac{-\sqrt{2}}{2}\right)\right)$ without calculator

This is a question from the practice exercises of Barron's AP Calculus. The directions state that I cannot use a calculator. (No trig tables I think, since those are not allowed in the exam) So, ...
2
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2answers
182 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
2
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2answers
67 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
2
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1answer
60 views

Equilateral Triangle Problem With Trig

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle P. The distance $|AP|=3 cm, |BP|=4 cm, |CP|=5 cm.$ What is the area of the ...
2
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1answer
828 views

$\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $?

How to prove, if $\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $ ? I am not familiar with trignometric identities. So any help will be appreciated. Thanks in Advance.
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2answers
739 views

Calculate coordinate of any point on triangle in 3D plane

I am really stuck and can't find right way to write a formula(s) that will calculate Z coordinate of point on triangle plane in 3D plane. I know all coordinates of triangle points ( Ax, Ay, Az, Bx, ...
2
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2answers
129 views

How to find the value of $4\cos(\frac{\pi}{26})+\tan(\frac{2\pi}{13})$

I have found in wolfram alpha that $\displaystyle 4\cos\left(\frac{\pi}{26}\right)+\tan\left(\frac{2\pi}{13}\right)=\sqrt{13+2\sqrt{13}}$. How to prove this identity ? Thank you.
2
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4answers
173 views

Finding $\lim_{x \to 0} \frac {a\sin bx -b\sin ax}{x^2 \sin ax}$ witouth L'Hopital, what is my mistake?

I was working on this question. $\lim_{x \to 0} \dfrac {a\sin bx -b\sin ax}{x^2 \sin ax}$ $\lim_{x \to 0} \dfrac {1}{x^2} \cdot \lim_{x \to 0} \dfrac { \frac {1}{abx}}{\frac {1}{abx}} \cdot \dfrac ...
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0answers
92 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
2
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1answer
549 views

Is sine of angles greater than 90 degrees a convention?

The sine function is defined as the opposite side of the angle in question over the hypotenuse of the $90^\circ$ triangle. $$\sin(â) = \frac{\text{opposite side}}{\text{hypotenuse}} ...
2
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1answer
127 views

$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$

Prove that (not use induction) $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) ...
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6answers
2k views

why is the square of this matrix with sin and cos equal to the identity matrix?

I have a question about why the square of the matrix Q, below, is equal to the identity matrix. Q = cos X -sin X sin X cos X My knowledge of ...
2
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3answers
104 views

If $p =\frac{4\sin\theta \cos\theta}{\sin\theta +\cos\theta}$ Find the value of $\frac{p+2\sin\theta}{p-2\sin\theta}$

Problem : If $\displaystyle p =\frac{4\sin\theta\cos\theta}{\sin\theta +\cos\theta}$, find the value of $\displaystyle \frac{p+2\sin\theta}{p-2\sin\theta} + \frac{p+2\cos\theta}{p-2\cos\theta}$. ...
2
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1answer
86 views

find parameter for maximize area

suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are $A(cos(3-t),sin(3-t))$ $B(cos(t),sin(t))$ ...
2
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3answers
220 views

Confusing Trigonometry Problem

Lets say at an intersection the words "STOP HERE" are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle ...
2
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1answer
358 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
2
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1answer
235 views

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$ For $\cos \theta$ I get: $\frac{-1}{k}, (as \theta$ is obtuse, ...
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1answer
125 views

Inverse trig question?

Hello everyone I have a question about trig. How would I solve the following. $$\tan\left(2\arcsin(4/5)+\arccos(12/13)\right)=\frac{253}{204}$$ Please help.
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1answer
153 views

A pseudo Fejér-Jackson inequality problem

$x\in (0,\pi)$ ,Prove that: \begin{align} \sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3 \end{align} the inequality holds for all integer $n$ I tried Fourier, or Dirichlet ...
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1answer
2k views

Proof of Osborne's Rule

Osborne's rule is described here. Firstly, am I right that only signs of terms in the form $\sin^{4n+2} \theta$, $n \in \mathbb{Z^+}$ have their signs switched (i.e. terms like $\sin^4 \theta$ simply ...
2
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1answer
3k 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 ...
2
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2answers
365 views

Angle for pointing at a certain point in 2d space

Recently, I have been programming a simple game. Very simple: There is a tank, and the cannon will aim at whatever position the mouse is at. Now lets talk about the cannon graphic. The cannon graphic ...
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2answers
165 views

Find the Outgoing Edge with the Smallest Angles, Given one Incident Edges and Multiple Outgoing Edges

I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know ...
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1answer
89 views

Proving $\int_0^{\pi } f(x) \, \mathrm{d}x = n\pi$

I've been asked to show $$ \displaystyle \int_{0}^{\pi} \dfrac{2(1+\cos x) - \cos((n-1)x) - \cos((n+1)x) - 2\cos nx}{1-\cos 2x} \ dx = n\pi $$ The integrand simplifies nicely to $$\frac{\cos nx - ...
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0answers
100 views

Riemann sum formulas for $\text{acos}(x)$, $\text{asin}(x)$ and $\text{atan}(x)$

In this post just another $\pi$ formula, I gave a kind of Riemann sum to compute the area of a quarter of circle based on a very simple geometric trick, and same reasoning can be used to compute any ...
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4answers
119 views

How to prove $\left(1+\frac{1}{\sin a}\right)\left(1+\frac{1}{\cos a}\right)\ge 3+2\sqrt{2}$? [closed]

Prove the inequality: $$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$
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0answers
52 views

How to find $\frac{x}{y}$?

Let $x=\displaystyle\prod_{n=1}^{180}\left(\cos{\left(\dfrac{n\pi}{180}\right)}+2\right)$ and $y=\displaystyle\sum_{n=0}^{89}\binom{180}{2n+1}\left(\dfrac{3}{4}\right)^n$. How to find $\frac{x}{y}$ ? ...
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5answers
166 views

Proving that $\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}$ [closed]

Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$
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1answer
69 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
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1answer
29 views

Replicating a cosine graph with sine, given transformations?

Say I have a cos function, $y = 1.2cos(0.503x) + 5$ And say I want to replicate it using sine, so expressing the above function in sine so that it gives "the same wave". How would I do this? I know ...
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3answers
76 views

Let $\theta=\frac{2 \pi}{67}$ consider the rotation matrix $A$. What is $A^{2010}$?

Let $\theta=\frac{2 \pi}{67}$. Consider the matrix $$A = \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin \theta& \cos \theta \end{pmatrix} $$ Then the matrix $A^{2010}$ is? My ...
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2answers
76 views

summation of a trigonometric series

How to evaluate $\tan^2(1) + \tan^2(3) + \tan^2(5) + \tan^2(7) + \ldots + \tan^2(89)$? Angles are given in degrees. I tried converting $\tan(89)$ as $\cot(1)$ and then tried combining $\tan(1)$ and ...
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8answers
89 views

Proving $\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$

I got this question from a paper but can't solve it and the question paper has no solutions section.How do you prove this? $$\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + ...
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1answer
73 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
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1answer
5k views

How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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2answers
117 views

Proof of trigonometric identity $\sin(A+B)=\sin A\cos B + \cos A\sin B$

All the proofs I've seen are geometrical, assuming that $A+B$ is less than $90$ degrees. How can you prove this identity for $A+B$ greater than $90$ degrees, or more generally, any arbitrary value?
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2answers
118 views

Trigonometric Arithmetic Progression

If $a$, $b$, $c$ are in arithmetic progression, prove that $$\cos A \cot\frac{A}{2} \qquad \cos B \cot \frac{B}{2} \qquad \cos C \cot\frac{C}{2}$$ are in arithmetic progression, too. Here, $a$, $b$, ...
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2answers
63 views

Trigonometric relation between sides and angles of a triangle

$$a \cdot \sin (B-C) +b \cdot \sin(C-A) +c \cdot \sin(A-B) =0$$ where $a, b, c$ are the sides of a triangle and $A, B, C$ are the angles of a triangle. No idea how to solve this problem.
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4answers
347 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
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3answers
1k views

How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$

How can I find the following product using elementary trigonometry? $$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$ I have tried using a substitution, but nothing has worked.
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2answers
291 views

Equation of a curve of a point moving inside a circle

Given circle $C_1$ of radius $r$ moving inside another circle $C_2$ of radius $R$ $(R\gt r)$ and tangent to this, I am in trouble to find the equation of the curve obtained by a fixed point on $C_1$ ...
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0answers
54 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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2answers
180 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
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3answers
349 views

Prove that $\sec^2{\theta}=(4xy)/(x+y)^2$ only when $x=y$

Show that the equation below is only possible when $x=y$ $$ \sec^2{\theta}=\frac{4xy}{(x+y)^2}$$ The only way I can think of doing this is by rewriting it as $$ ...
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0answers
61 views

Determining the location of a point in a triangle under the given constraints

ABC is a triangle with AC = 1, AB = c/b and BC = a/b. Q is a variable point on AC such that CQ = x and QA = 1 – x. The perpendiculars from A and C to BQ (extended if necessary) are $d_2$ and $d_1$ ...
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1answer
484 views

$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$ given that the sum of the angles is $\pi$

Question: Find the least real value of $M$ such that the following inequality holds: $$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$$ Given that $\alpha, \beta, \gamma \in \mathbb{R}^+$, ...
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3answers
6k views

Using De Moivre's Theorem to prove $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$ trig identity

I am stuck on trying to prove a trig identity using De Moivre's theorem. I have to prove, $$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$$ I am not sure where to even start, I broke the LHS down ...
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1answer
58 views

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...