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

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23 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
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108 views
+100

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
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0answers
14 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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2answers
15 views

Evaluating trig functions for a point that passes through…

I have the question "Evaluate the trig functions for angle a in standard position whose terminal side passes through (3, 4): Sec a, csc a, and cot a. For cot a the answer given is 3/4, which makes ...
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1answer
30 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
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2answers
289 views

'Rational' solutions of sine

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q$$ Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$? ...
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1answer
38 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
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1answer
54 views

Calculationg the angle of a triangle

I am trying to find a specified angle of a triangle. In triangle $ABC$, $\angle A = 20^\circ$. $D$ and $E$ are points on $AB$ and $AC$, where $AB=AC$. $\angle EBC = 50^\circ$ and $\angle DCB = ...
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1answer
16 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
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1answer
27 views

Help With Solving Trigonometric equations

$(\sin x)^2 - 5\sin x \cos x=0$ What would be the first atep to solve this. I normally get the equation into a quadratic one but I cannot seem to spot the first step here. What I mean by $(\sin ...
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1answer
21 views

Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
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1answer
72 views

Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
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3answers
41 views

Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
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4answers
176 views

Complex numbers - Exponential numbers - Proof

Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6. For this problem I am stumped...how should I begin? Also there's a ...
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1answer
43 views

How to solve: y'' + 9y = sin(3t)

I need to find the particular solution to the equation: $$y'' + 9y = \sin(3t)$$ I thought we were looking for a trigonometric forcing term on the form: $$y = a\cdot\cos(3t) + b\cdot\sin(3t)$$ But ...
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1answer
494 views

Determine sin cos and tan from slope? NON CALCULator

I am used to finding this by drawing the triangle and knowing the angle measurement. With the angle measurement i can find sin cos and tan. But i dont have angle all I have is line. I have put a ...
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5answers
79 views

Solving a trigonometric equation: $2 \sin(3a)=\sqrt{2}$

I have the following equation : $2 \sin(3a)=\sqrt{2}$ Not sure how to solve it (Because it's a transformed sin function, meaning 6 solution with 3 cycles in $2\pi$) after a moment I finally found ...
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2answers
68 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
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5answers
716 views

Solving the equation $\sin t = -\sqrt{2}/2$

Solving the equation $$ \sin t = -\frac{\sqrt{2} }{2} .$$ I know the solution is $1.25$ and $1.75$, but I do not know how to get there. An explanation would be GREATLY appreciated, thanks!
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1answer
13 views

Calculate perimeter of rhomboid

I am trying to solve the following problem but I got stuck In a rhomboid with an area of $48 \space cm^2$, the major diagonal is $4$ cm shorter than the double of the minor diagonal. Calculate the ...
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3answers
27 views

Find base of isosceles triangle with side length and angle

I would like to calculate the length of the side in red on the image. I tried the Law of cosines, but maybe i haven't applied the formula right, because for a side "a" and "b" of size 64 and a angle ...
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1answer
39 views

Proving standard properties of sine and cosine defined by their power series

Definition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n ...
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1answer
46 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
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4answers
38 views

Prove the inequalities without calculating the integrals

$$ \int_{0}^{\frac{\pi}{2}} \sin^4x dx \le \int_{0}^{\frac{\pi}{2}} \sin^3xdx$$ I have tried to define 2 functions $ f, g:[0, \frac{\pi}{2}] \rightarrow \mathbb{R}$ and say that $ f(x) = \sin^4x$ ...
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2answers
86 views

Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$. I know that these numbers add up, but is there any ...
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3answers
238 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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1answer
50 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [on hold]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
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5answers
246 views

Derivation of the limit $\sin x/x$

Reference: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? I think that all proofs above are beautiful but I cannot see why we have to complicate it. We know that for small values of $x$ that ...
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1answer
26 views

Square Wave Intuition

As I understand it, a square wave can be produced as follows: $$y = \cases{ 1 & \text{if } \sin(x) > 0\cr 0 & \text{if }\sin(x) = 0\cr -1 & \text{if } \sin(x) < 0} $$ What I'm ...
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37 views
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1answer
30 views

Find all angles that satisfy $6\cos^2(x)+5\cos(x)-6=0$ [on hold]

Find all angles that satisfy: $$6\cos^2(x)+5\cos(x)-6=0.$$
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1answer
25 views

angle $0$ to $2\pi$ between two 3Dvectors

Ok this is for a computer game I'm learning to program with. How do you find angle between two normalized 3D vectors so that you get the resulting angle in the range $[0,2\pi]$ or $[-\pi,\pi]$? Using ...
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1answer
110 views

Measuring Earth Radius using distance between two tall buildings

Measuring Earth's radius using two tall building heights H and h with a ground separation s and also when the building tops are collinear with the horizon: Concluding result of concurrent thread " ...
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2answers
98 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
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1answer
122 views

Is there any identity for $\sum_{k=0}^{n-1}\tan(x+ka) $??

I found this series $$ \sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right) $$ but it's not what I need.
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3answers
94 views

How prove $\sin \left( \alpha+\frac{\pi }{n} \right) \cdots \sin \left( \alpha+\frac{n\pi }{n} \right) =-\frac{\sin n\alpha}{2^{n-1}}$?

How prove $$\prod_{k=1}^{n}\sin \left( \alpha+\frac{\pi k }{n}\right) =-\frac{\sin n\alpha}{2^{n-1}}$$ for $n \in N$?
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1answer
51 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
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2answers
24 views

Solving a Cartesian and parametric equation at a intersection.

A curve C has parametric equations: $x=4cos(2t)$ and $y=3sin(t)$ $-\frac{\pi}{2} < t < \frac{\pi}{2}$ The normal of a point A$(2,1.5)$ on curve C has the equation $6y-16x+23=0$ The curve and ...
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1answer
58 views

Why is arcsin represented with the ^(-1) notation?

So in trigonometry, we have sin, secant (which is one over sin) and arcisn. Why is arcsin sometimes represented with sin^-1? sin^2 means sin to the second power, but sin^-1 explicitly does not mean ...
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1answer
26 views

How to divide trigonometric ratios using identities?

$$\frac{1-\tan^2x}{1+\tan^2x}$$ We know: $$\frac{1-\frac{\sin^2x}{\cos^2x}}{1+\frac{\sin^2x}{\cos^2x}}$$ Now what? Flip denominator and times numerator? Which equals ??? Please help - Thanks
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2answers
29 views

Solve trigonometric equation $\sin(2t) = −\frac{\sqrt{2}}{2}$ on interval $[0, 2 \pi]$

Solve the following equation on interval $[0, 2 \pi]$: $$\sin(2t) = −\dfrac{\sqrt{2}}{2}$$ I got $t=\left\{\dfrac{9\pi}{8},\dfrac{15\pi}{8}\right\}$, but website for math assignment said that it is ...
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1answer
16 views

Find the area of trapezium given certain angles and length of diagonal

In the trapezium $MNOP$, $MP$ is the major base and $NO$ is the minor base. Knowing that the angle $P$ is $58° 15'$, the angle $OMP$ is $21° 45''$, and the diagonal $OM$ is of $6.5$ cm, calculate the ...
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1answer
31 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
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2answers
35 views

Find Coefficients from already fourier function

Hello I have this function and I'm asked 1.Find the period for $f(t)$ 2.Find the coefficients $a_n$ and $b_n$ $$f(t)=2(cos(2t+\frac{\pi}{4})-sin(6t-\frac{\pi}{2}))$$ I know that the period for ...
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2answers
215 views

Evaluation of the integral $\int_{-6}^{-3}\frac{\sqrt{x^2-9}}{x}$

How to evaluate the following integral? I have tried the following things but I have no idea to continue after the last step. Moverover, the integral seems wrong when compared with the ans from ...
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3answers
80 views

Evaluation of a general trigonometric integral

How can I evaluate the integral $$\int\sin^k(x)\ dx$$ in which I don't know if $k$ is an even or an odd number?
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346 views

Solving a simple system of equations

Given the simultaneous equations $$A\cos{(\sqrt{\lambda}\pi)} + B\sin{(\sqrt{\lambda}\pi)} = 0$$ $$A\cos{(2\sqrt{\lambda}\pi)}+B\sin{(2\sqrt{\lambda}\pi)} = 0$$ We want to show this has not trivial ...
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0answers
24 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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1answer
29 views

Find sides and height of isosceles trapezium given information about its diagonals

In an isosceles trapezium the diagonals cut at a point $O$ which divides them in two segments of $3$ cm and $7$ cm. If one of the angles formed between them is of $120°$, find the measures of the ...
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0answers
30 views

What is the maximum area of a trapezium with 3 known sides and unknown angles. [on hold]

The Question: A major company in your city has both new equipment capable of making guttering in the shape of an open top trapezium. The sheet metal used is 22 cm wide and bent such that the base s ...