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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1answer
53 views

Evaluating sinusoid at Chebyshev points

Suppose I have a sinusoid $f(t) = A \cos(\omega t + \theta)$ and I want to evaluate it at Chebyshev points of the second kind ($\cos(\frac{2 \pi i}{N}), 0 \le i \le N, i \in \mathbb{Z}$), and then ...
3
votes
0answers
70 views

When $\cos{n^{\circ}}$ can be expressed in real radicals?

So, the question is: $\cos{n^{\circ}}$ can be expressed in real radicals iff $3 \mid n$? Is it true? The first part is easy: if $3 \mid n$ we can express it, because ...
0
votes
2answers
623 views

Distance between centroid and incenter in a right-angled isosceles triangle

Let ABC be a right-angled isosceles triangle where AB = BC = a. Assume that C is its centroid and I is its incenter. Find, in terms of a, the distance between C and I. Answer : $CI= \frac{{a \cdot ...
3
votes
1answer
94 views

Brouwer's fixed-point theorem and iterative convergence of a composition of circular functions

let $\psi:[0,1]\times \{0,1\} \rightarrow [0,1]$ be defined by: $$ \psi(x,\beta) = \beta \cos x + (1-\beta) \sin x $$ define $B_n$ as the set of $2^n$ binary strings $b=b_0b_1\dots b_{n-1}$ where ...
4
votes
1answer
162 views

convergence of the iterated cosine

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying: $$\cos \alpha = \alpha$$ it is also evident (empirically) that ...
3
votes
3answers
1k views

What is arcsec (-2)?

The question asks to solve $\operatorname{arcsec}(-2) $. Options are a) π/3 b) −(π/3) ...
6
votes
0answers
130 views

is the unique solution of $\cos t = t$ a transcendental number?

let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$ for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were ...
1
vote
1answer
115 views

hitting a pool ball at exactly 90 degrees

So a friend and I had an argument and I thought I might get some unbiased proof here. Is it possible to hit a pool ball with another pool ball at exactly 90 degrees. Ie if I had a white ball could I ...
0
votes
1answer
216 views

Given three points, find the arc length of a section between two intersecting lines.

I have three points, one being the center, and the other two are end points on a line drawn to the center. I need an equation that provides $\Theta$. In this drawing $(x_1, y_1)$ is the center.
2
votes
5answers
834 views

$\int\sin^6(x)\cos(x)dx$.

$$\int\sin^6(x)\cos(x)\,dx.$$ Apparently... $$\int\sin^6(x)\cos(x)\,dx= \int\sin^6(x)\,d\sin(x)x = \frac{1}{7}\sin^7(x)+c.$$ Can somebody explain the second equality? Shouldn't there be a $dx$ ...
1
vote
1answer
122 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
1
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3answers
397 views

$\tan2x$ in terms of $\cos x$ alone

This is not exactly a homework question but something I was trying to do to get my basics back on track. I wanted to find $\tan2x$ in terms of $\cos x$ alone. I was able to do it in terms of $\sin x$ ...
2
votes
2answers
591 views

Determine the smallest positive value of x(in degrees) for which: $\tan(x+100) = \tan(x+50)\tan (x)\tan(x-50)$

Determine the smallest positive value of x(in degrees) for which: $\tan(x+100) = \tan(x+50)\tan (x)\tan(x-50)$ I tried to apply the formula of $\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A \tan B}$ ...
4
votes
1answer
122 views

Some weird system of equations.

How do you solve this type of system of equations? The unsubscripted variables are to be found. $A^2+B^2={C_1}^2$ $C^2+D^2={C_2}^2$ $E^2+F^2={C_3}^2$ $(A+C)^2+(B+D)^2={C_4}^2$ ...
1
vote
1answer
40 views

increase in diagonal

I am looking at the following figure, showing the deformation of a triangular element from time $t$ to time $t+dt$. Note that the $t$ on the left leg of the isosceles triangle simply denotes our ...
1
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3answers
115 views

A resource for Trigonometric Inequalities

I'm looking for a good and detailed guide for trigonometric inequalities in pdf if possible. Any recommended resources?
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3answers
61 views

Plane Geometry problem

I came across this problem in a mathematics-related facebook group. Could anyone advise on the solution to it(i.e. hints only)? Thank you.
3
votes
3answers
143 views

Prove that $A + B = C$

I drew the diagram here I honestly do not see how $A$ and $B$ could possibly equal $C$.
1
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5answers
85 views

how do i write $y = 2\sin2(x + \frac{\pi}{4}) - \cos2(x + \frac{\pi}{4} )$ in format $y = A\sin(kx) + B\cos(kx)$

The problem is the double angles. I tried to simplify them and change them around but no luck, $$\begin{align} y&=A\sin(kx)+B\cos(ky)\\ y&=2\sin2(x+\pi/4)-\cos2(x+\pi/4)\\ ...
2
votes
1answer
222 views

How do I put in words that $2\cos^2 x +\sin x-1$ cannot be factored?

I know that $2\cos^2 x +\sin x-1$ can't be factored, but I don't really know how to explain it. The best explanation I can come up with is "it would be like factoring $2x^2 + y -1$; it just doesn't ...
4
votes
4answers
127 views

How do I verify $\frac{(\sin x - \cos x + 1)}{(\sin x + \cos x-1)}=\frac{\sin x + 1}{\cos x}$?

I need to verify this trigonometric identity for an assignment:$$\frac{(\sin x - \cos x + 1)}{(\sin x + \cos x-1)}=\frac{\sin x + 1}{\cos x}$$ I've tried a few different approaches, but I end up ...
4
votes
4answers
88 views

Differential equation trouble

I am trying to solve the following differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=2(2x+y)^2$$ If we make the substitution $z=2x+y$, then we get: ...
2
votes
3answers
382 views

Limit n tends to infinity

How can i solve this: $$ \lim_{n\to\infty} \cos(1)\cos(0.5)\cos(0.25)\ldots \cos(1/2^n) $$ I tried using comlex numbers and logarithms but did'nt work out.Can anyone help please.
2
votes
1answer
417 views

Pre-College Maths Textbooks

I am a high school student searching for some mathematics books covering material all the way up to, but not including, college level mathematics. I have already read Gelfand's books and Lang's Basic ...
2
votes
2answers
130 views

Closed form for $\frac2\pi\int_0^{\frac{\pi}{2}}\sqrt{t^2+r^2-2tr\cos(\alpha)}\,\mathrm d\alpha$

Does the following integration have an explicit solution? Thanks a ton for the help. Best, Ruinan $$\dfrac2\pi\int_0^{\pi/2}\sqrt{t^2+r^2-2tr\cos(\alpha)}\,\mathrm d\alpha$$
4
votes
8answers
1k views

How to solve $\sin x +\cos x = 1$?

No matter how I do it, I always end up with $x = 0, 90, 270$ and $360$. All of those except $270$ is right, but I can't quite figure out how to get the $270$ degrees out of the answer. I've tried ...
2
votes
4answers
1k views

Can I use the quadratic formula when there is no constant term?

I was wondering if it's erroneous to use the quadratic formula on a quadratic equation where there is no constant term. What I figured I'd try was to just assome the constant term is +0. I was doing ...
2
votes
2answers
84 views

Approximate range of sum of two trig functions.

Find the approximate range of the function y = 2 sin (6x) + sin (4x). My initial reasoning is that sin of anything maxes out at 1, so this function can be rewritten as y = 2 (1) + 1 The maximum ...
2
votes
2answers
46 views

Need Sine form of Cotangent equation

$$(b^2 - c^2)\cot A + (c^2 - a^2)\cot B + (a^2 -b^2)\cot C=0$$ I want this equation to be in the Sine form. Please help me with steps. Thanks a lot
1
vote
1answer
78 views

Why doesn't this work?

http://imgur.com/g5yhHLK I'm trying to find the sum of $\cos(\frac{\pi}{k}) + \cos(\frac{2\pi}{k}) + \cos(\frac{3\pi}{k}) + ... + \cos(\frac{n\pi}{k})$ I multiplied it by $2\sin(\frac{\pi}{k})$ and ...
1
vote
1answer
91 views

Write function from polar to rectangular coordinates.

I need to write this functions in rectangular coordinates: $$f(r,\theta)=r^{2k+5}\cos5\theta$$ $$g(r,\theta)=r^{2k+5}\cos5\theta$$ Of course the radius is very easy to convert to $x$ and $y$. The ...
5
votes
5answers
397 views

Trigonometry equation $\sin(x)+\cos(x)-\tan(x)=0.4$

There's some way to find $x$ here ? $$\sin(x)+\cos(x)-\tan(x)=0.4$$
4
votes
2answers
125 views

Prove that $|\sin n|+|\sin (n+1)| > 2\sin(1/2)$ for all $n\in \mathbb N$

Show that $$|\sin{(n)}|+|\sin{(n+1)}|>2\sin{\dfrac{1}{2}},n \ge 1,n\in \mathbb N$$ My try: let $$F(n)=|\sin{(n)}|+|\sin{(n+1)}|$$ then ...
8
votes
2answers
219 views

calculation of $\int_{0}^{1}\tan^{-1}(1-x+x^2)dx$

Calculation of $\displaystyle \int_{0}^{1}\tan^{-1}(1-x+x^2)dx$ $\bf{False\; Try}::$ Let $1-x+x^2=t$, Then $\displaystyle \left(2x-1\right)dx = dt\Rightarrow dx = \frac{1}{(2x-1)}dt$ Now Changing ...
3
votes
1answer
175 views

How do I solve $\sin^2 x=\cos x$?

I'm trying to solve a trigonometric equation, but I'm a bit stuck. The equation is this: $\sin^2 x = \cos x$ So far what I've done looks like this: $\sin^2 x - \cos x = 0$ $ (1 - \cos^2 x) - ...
4
votes
1answer
150 views

Convergence of $\sum\frac{\tan(nz)}{n^2}$ to an analytic function…what if $z\in \mathbb{R}$?

For which values of $z$ does $$\sum_{n=1}^\infty \frac{\tan(nz)}{n^2}$$ converge? For which values of $z$ is the limiting function analytic? One can show, as in this answer, that ...
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3answers
103 views

$\sin (A-B) = \pm\frac{1}{3}.$ if…

please help me, how can I approach. I can not understand what to do. How can I start. Help me. If $\sqrt{2}\cos A =\cos B +\cos^3 B$ and $\sqrt{2} \sin A = \sin B - \sin^3 B$ show that $\sin (A-B) ...
0
votes
2answers
60 views

A simple question on inverse trigonometry

$$ \arctan \frac{2x+1}{ \sqrt{3}} + \arctan \frac{2y+1}{ \sqrt{3}} = ??$$ Isnt the formula $\arctan x + \arctan y = \arctan \frac{x+y}{1-xy}$ ? My answer doesn't match with the textbook's Thank ...
1
vote
1answer
42 views

Are there specific terms for trigonometric functions raised to a power?

Related to my other question, asking for a Book on higher-power trigonometric equation simplification techniques, I am interested to learn if there are specific terms for trigonometric functions that ...
4
votes
2answers
150 views

Area of an equilateral triangle divided by three lines

An equilateral triangle is divided by three straight lines into seven regions whose areas are shown in the image below. Find the area of the triangle. How to solve this problem ?
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1answer
65 views

What did griffiths do here? Trig Identity for buffons needle

I'm on the very last part of the equation for buffons needle. I think it's a trig identity, but I can't find it. Either way, I can give more info if needed but it looks like this.
0
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1answer
37 views

Question o Trigonometrical equation

I'm know confused by an equation involving $tan$ Given: $tan\bigl(x - \dfrac{\pi}{4}) \le 1$ for $-\pi < x < \pi$. Adjusting the bound for $x - \dfrac{\pi}{4}$ it becomes $\dfrac{-5\pi}{4} ...
1
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4answers
245 views

Upon multiplying $\cos(20^\circ)\cos(40^\circ)\cos(80^\circ)$ by the sine of a certain angle, it gets reduced. What is that angle?

So if $P = \cos(20^\circ)\cos(40^\circ)\cos(80^\circ)$, I can multiply $P$ by $\sin(X)$ so that the entire expression reduces to something manageable. I then take the simplified product and divide it ...
0
votes
1answer
180 views

Converting from spherical coordinates to cartesian around arbitrary vector $N$

So if I'm given an arbitrary unit vector $N$ and another vector $V$ defined in spherical coordinates $\theta$ (polar angle between $N$ and $V$) and $\phi$ (azimuthal angle) and $r = 1$. How do I ...
6
votes
1answer
2k views

Intersection of Trig Functions

The questions asks to find the intersections of $$f(x) = 2 \sin(x-7) + 6$$ and $$g(x) = \cos(2x-10) + 8$$ within the interval $[6,14]$. So my general strategy was, 1) equate the functions, 2) get ...
2
votes
1answer
189 views

A Nonlinear System of Trigonometric Equations

folks! I am getting the following system of trigonometric equations which has come as a product of my research. But, I have tried a lot with no success, to solve this system. Can anyone help? Thank ...
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vote
2answers
80 views

2.5d game render math problem

So I'm making a star Ship bridge game where the game is rendered using a 2-D Cartesian grid for positioning logic. The player has only the attributes of position and an arbitrary look-at angle ...
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3answers
78 views

Doubts in Trigonometrical Inequalities

I'm now studying Trigonometrical Inequalities, and I've just got struck when I have modified arguments to my trigonometrical functions, for example: $\sqrt{2} - 2\sin\left(x - \dfrac{\pi}{3} \right) ...
3
votes
3answers
118 views

$\cos^2\frac{1}{2}(\alpha-\beta)=\frac{3}{4}$ if…

Help please: If $\sin\alpha+\sin\beta= \sqrt{3} (\cos\beta-\cos\alpha)$ then show that $\cos^2\frac{1}{2}(\alpha-\beta)=\frac{3}{4}$ ...
2
votes
0answers
51 views

Book on higher-power trigonometric equation simplification techniques

Recently, I have become fascinated about learning techniques for simplifying experimentally derived trigonometric mathematical models that are raised to higher powers. Are there any good references ...