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

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Solving $q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$ for $B_x$

I'm a bit rusty and am having trouble using Trig Identities to solve for $B_x$. Can someone show me how to do this? $$q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$$ I want to solve for $B_x$ ...
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54 views

Find: $\sin\left(\frac{2}{\arcsin((x + 4)/5)}\right)$

Find: $$\sin\left(\frac{2}{\arcsin(\frac{x+4}{5})}\right)$$ I know: $$\sin(\arcsin(x)) = x$$ I somehow did something that did get this correct.: $$\sin(2t) = 1 - 2\sin^2(t)$$ So then we see: ...
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50 views

Find Laplace Transform of trigonometric function using unit step function and t-shifting. (5.3-40)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=sin(t)u(t-\frac{\pi}{2})$$ My solution: Use the following corollary from the second shifting theorem ...
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36 views

Hockey pucks and parameters

There is one hockey puck with a diameter of 3 inches. The puck is spinning around its center at a speed of 3 counterclockwise rotations per second. At the center, the puck is traveling at a speed of ...
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54 views

Calculate vertical lines intersection hexagon at regular interval

I would like to calculate the total size of vertical lines that dissect an hexagon regular, like the one on the image. I would like to know the internal size of the blue lines inside the hexagon, ...
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36 views

What is the best trigonometry book available free?

I am not a rich person but I really want to have a look on the trigonometry book
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18 views

From a light house L two ships P and Q are observed in direction South West and $5^{\circ}$ East of South respectively. At same time Q…

Question : From a light house L two ships P and Q are observed in direction South West and $5^{\circ}$ East of South respectively. At same time Q is observed from point P in South East direction of ...
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67 views

Why is the “i” disappearing?

The task is: Find the argument in its simplest form. $$(\sin(x) +i(1-\cos(x)))^2$$ where $x$ is an acute angle. I multiplied out the equation and let alpha be the required argument, then ...
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29 views

How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...
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31 views

Conditional inequality

Let x,y,z be positive reals with $xy+yz+zx=1$. Prove the inequality $$\sum_{cyc(x,y,z)}\frac {2x(1-x^2)}{(1+x^2)^2} \le \sum_{cyc(x,y,z)} \frac x{1+x^2}.$$ I substituted $x=tan\frac{\theta}2, ...
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21 views

Question about Chebyshev Polynomials in Beardon

I happen to be reading through Beardon's book, Iteration of Rational Functions, and I have come across a statement I don't quite believe. He uses it a little later on, so I'm concerned with clearing ...
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47 views

How do I find the angle measurement on a triangle that has one curved side?

I have tried taking this from a circle and measuring the angles as if the width and the height are the quarter of a circle however it is not measuring correctly. I have looked at it as if it is the ...
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49 views

What function has a 3D graph that will look like a spiral into a singularity?

I am trying to draw text spiraling into a black hole, from a more interesting slightly off-orthogonal viewpoint. I think a function that defines a black hole/singularity surface might look something ...
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63 views

Did Wolfram Alpha make this term up?

So I'm currently taking a Trigonometry class and I'm using the iOS version of the Wolfram-Alpha app to assist me with some of the more mundane calculations (Pythagorean Theorem, etc) and I've come ...
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71 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
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30 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
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48 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
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47 views

Find the length of the longer diagonal on a trapezium with only 2 sides stated.

Im at a loss here, i know i have to divide the trapezium, but im still not sure which calculation is relevant to it then. Thanks in advance.
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23 views

Calculate angle of view from 2D image

I want to calculate the angle of view (or the field of view) from a photograph, without knowing anything about the camera, as to use that information in a 3D environment. I have to use trigonometry ...
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113 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
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26 views

Formula for cos((2n+1)x) as polynomial of cos x

I am looking for a formula of cos((2n+1)x) that is polynomial of cos(x). For example, Is it known for any n?
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27 views

Practical determinations of trigonometric identities

I am looking for articles, or any reference, that detail practical determinations of trigonometric identities, with particular emphasis on trigonometric functions raised to the power of 3 or higher. ...
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90 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
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106 views

The geometry of a spiral made of adjacent right triangles

In the above figure (not sure if you can see it clearly or not), while using the old standard technique of plotting irrational numbers on number line, I saw this property. If we go on plotting ...
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81 views

Calculating point on sphere surface where sun reflection to a target point occurs

Imagine a mirror sphere at position O with radius R, and a target point at position P, at distance d from the sphere origin. There is an unknown point X on the surface of the sphere, where the light ...
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133 views

Trigonometric curiosity

How prove this $$-\tan\frac{10\pi}{41}+4\left(\sin\frac{2\pi}{41}+\sin\frac{4\pi}{41}+\sin\frac{12\pi}{41}+\sin\frac{20\pi}{41}-\sin\frac{26\pi}{41}-\sin \frac{30\pi}{41}\right)= ...
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42 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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40 views

How to find the period of a exponential function? $5\cdot(-1)^k$

Currently in class we are learning Euler's formula and how it relates to imaginary numbers, exponentials, and trig functions. I know the left hand of the equation is $e^{jwt}$ and I could easily use ...
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16 views

Nonnegativity on a special domain entails nonnegativity on the whole plane

Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ...
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61 views

Calculate area of this figure

I have an homework assignment where I have to calculate area of the figure underneath. I used the following formula to calculate the result $\frac {130 \cdot 55 \cdot sin35}{2} = 4188 m$ and then ...
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54 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
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51 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
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85 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
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21 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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80 views

Period of trigonometric function

What is the period of $$\frac{7\sin x + 5\cos x}{7\sin{2x} + 11\cos x}$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint!! Thanks.
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68 views

Calculate a point on a geodesic line on an ellipsoid

I have a problem which i don't understand how to achieve. Maybe someone could sheed some light on it. Have a look at this picture: What I try to achieve is to determine the point D on the geodesic ...
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67 views

Trig equation that fits the plot points (octagonal pyramid)

I'm looking for an equation that satisfies these conditions: Input 90 degrees, result is 90 degrees Input 45 degrees, result is 60 degrees Input 0 degrees, result is 45 degrees For an input value ...
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50 views

Is my answer correct, or textbook correct?

My answer is 9.45, textbook is 4.7 I did cos 38 = x / 12, 12 cos 38 = x 9.45 = x ty in advance
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49 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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43 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
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31 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
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Horizontal axis for reference angles

Why we always take the horizontal axis for reference angles? Is it by convention? Could it have been the y-axis? What advantages do we gain from taking the horizontal axis?
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Is this solution correct

Given $A+B=\frac{\pi}{4}$, find $(1+\tan A)(1+\tan B)$ My attempt: Since $\tan(A+B)=1=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan (B)}$, therefore $\tan(A)+\tan(B)+\tan(A)\tan(B)=1$, therefore, ...
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Weighted sum of $\cos(nx)$ series

This is a follow up question to Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$ I am looking ...
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39 views

Weird inequality answer, truncate or round?

When arriving at the final answer for a double inequality question, it appears that my text book has truncated one part and rounded the other. Is there some weird inequality rule that I don't know ...
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85 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
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64 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
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29 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
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68 views

Solving an equation with $\arccos(x)$ and $\sin(\arccos(x))$

I want to solve this equation, determining y (all others letters are constants) : $$2 \arccos(3+(1.6y-80)/R) - \sin(2\arccos(3+(1.6y-80)/R)) = 2π(1-P)$$ I've try to use some automatic solvers but ...