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
67 views

Construct sequence $z_n$ with $|z_n|\to\infty$ s.t. $\tan(z_n)\to c$

If we are given a constant $c\in $ the extended complex plane, how can we construct a sequence of complex numbers $z_n$ with $|z_n|\to\infty$ such that the corresponding sequence $\tan(z_n)\to c$? I ...
1
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1answer
484 views

Given the width and rotation of rectangle, calculate it's unrotated width

In web graphics if you ask for the width property of a rectangle you'll get back the horizontal measurement of the rectangle at that time rather than it's starting width. So a rectangle that is longer ...
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2answers
237 views

How do I get $\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$?

How do I get: $$\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$$
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1answer
84 views

Behaviour of $\tan$

What happens to $\tan(u+iv)$ as $u^2+v^2\to \infty$ via a path where $v\neq 0$ and $u,v\in \mathbb R$? How can I tell?
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1answer
469 views

Why $\arccos(\frac{1}{3})$ is an irrational number?

I was reading the following question. It is a very nice question with a very nice answer! I would like to know why $\arccos(\frac{1}{3})$ is an irrational number.
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2answers
268 views

What does $\sin^{2k}\theta+\cos^{2k}\theta=$?

What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to? Besides Mathematical Induction,more solutions are desired.
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1answer
437 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
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1answer
646 views

Solve x = sin(t) for t

How can I solve: $(\frac{x}{16})^{\frac{1}{3}} = \sin(t)$ for t?
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3answers
538 views

unit circle, derive number for any degree, cosinus and sinus

$\sin(90°)= \sin(\frac{1}{2}\pi)= 0$ $\cos(90°)= \cos(\frac{1}{2}\pi)= 1$ $\sin(60°)= \sin(\frac{1}{3}\pi)=\frac{\sqrt{3}}{2}$ $\cos(60°)= \cos(\frac{1}{3}\pi)=\frac{1}{2} $ $\sin(45°)= ...
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2answers
446 views

Solving $\arctan(a) + \arctan(b) + \arctan(c) = \pi$ for $0 < a < b < c < 10$

This is a trigonometry math contest problem. Which ordered triple of numbers $(a,b,c)$ with $0 < a < b < c < 10$ satisfies the equation $$\arctan(a) + \arctan(b) + \arctan(c) = ...
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1answer
3k views

How do you find the value of theta in this example?

My problem says to find the measure of each acute angle $\theta$ to the nearest degree. $$\large\cos\theta = 0.2249$$
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2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
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0answers
136 views

How can this trigonometrics equation be solved exactly, if possible?

I was working on an approximation for the sine function, in which I needed to calculate the maximum error to work on a compensation polynomial. My approximation was this: $$f(x) = \frac {4} {\pi^2} x ...
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5answers
771 views

Number of Solutions of $3\cos^2(x)+\cos(x)-2=0$

I'm trying to figure out how many solutions there are for $$3\cos^2(x)+\cos(x)-2=0.$$ I can come up with at least two solutions I believe are correct, but I'm not sure if there is a third.
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3answers
115 views

Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?

For some reason, I cannot see a clever way to solve this (I know the way of doing it like in Wolframalapha) but I am pretty sure there is a double angle identity to crack this puzzle. Could someone ...
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1answer
2k views

How to calculate relative degree changes in 0 to 360.

I'm working on a project that measures wind direction and I'm stuck on this what appears to be a simple degree problem. Example: Lets say I'm a compass (0' .. 360' ) now I'm pointing due north 0' , ...
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1answer
97 views

Basic Trig problem

This is a very basic question but that I need to know to solve a harder calculus question. How do I solve for $x$, for the problem $\tan(x) = \sqrt{3}$?
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1answer
93 views

I need to factor this function so it is entirely dependent on x- semicircle displacement.

In the attached image are three functions. The first is a displacement function which takes angle $t$, and returns a radius. The third one is a semicircle and the second one is a semicircle with the ...
0
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2answers
755 views

Notation of inverse trigonometric functions and exponentiation [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ I have worked a bit on trigonometry today, and something strikes me as inconsistent. In the book, the notation for the inverse sine ...
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2answers
355 views

How to find $\int_{\frac{\pi}{2}}^{\frac{\pi}{4}}\cot^5x\,\csc^3xdx$

I stack about following problem... $\int_{\frac{\pi}{2}}^{\frac{\pi}{4}}\cot^5x\,\csc^3xdx$ I tried to change $\cot^5x=\frac{\cos^5x}{\sin^5x}$ I got ...
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4answers
160 views

How can I plot the point of $y=\sin(x)$?

I am taking an online trig course so I don't have the luxury of asking for help when I don't understand something. How can I plot $y=\sin(x)$?
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2answers
314 views

How to show this equation is true?

How can I show that this equation is true for all $x \in \mathbb{R}$? $$\sin^6x + \cos^6x = 1 - 3\sin^2x \ \cos^2x$$
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1answer
207 views

Question on trigonometric linear equations

Doing an exercise on complex analysis where it began by asking me to solve some equations for $u_x$ and $u_y$ I got stuck and looked up the answer. $$u_x\cos\theta+u_y\sin\theta=u_r,$$ ...
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1answer
170 views

Expressing the four roots of a particular quartic in terms of trigonometric functions

I know one root of the equation (eq.1), $x^4+ax^3+2x^2-ax+1 = 0$ is, $x_1 = \tan\big(\tfrac{1}{4}\arcsin(\tfrac{4}{a})\big)$ How to find the other three roots of eq.1 expressed similarly in terms ...
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2answers
1k views

What are functions used for?

When I say functions, I don't mean the trigonometric functions like $sin$, $cos$, and $tan$, I mean defined functions like $\large f(x) = 2x + 4$. Why is $\large f(x)$ used and why isn't a single ...
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3answers
114 views

Simplify $\sin^3{\left(\cot^{-1}{\left(x\right)}\right)}$

How can the following function such that no trigonometric functions are present: $\sin^3{\left(\cot^{-1}{\left(x\right)}\right)}$ Wolfram|Alpha shows the result as $\frac{1}{{\sqrt{x^2+1}}^3}$. ...
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2answers
328 views

Prove this trigonometric identity in quadrilateral

If $\alpha,\beta,\gamma,\delta$ are angles in quadrilateral different from $90^\circ$, prove the following: $$ ...
2
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3answers
271 views

Product of tangents

I was able to reduce an equation I have to: $$f(t) = \tan(\mu) \tan(\nu) - C = 0$$ where $\mu, \nu$ are linear functions of t and $C$ is a constant. Are there any identities for the product of ...
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1answer
233 views

polar graphs and investigation

I am new to polar graphs and I am trying to investigate some certain cases: What happens when you change the $b$ value to different positive integers in polar equations of the forms: ...
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8answers
3k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
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2answers
789 views

What are these specific trigonometric functions used for?

$\cos\theta$, $\sin\theta$, $\tan\theta$, $\csc\theta$, $\cot\theta$, $\sec\theta$ What are these all used for? Are they used to find the measure of a give angle provided the measurements of other ...
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1answer
190 views

If $\cos x = \frac{1}{\sqrt{5}}$, what is $\cos^{-1} x$?

Given that $\cos x = \frac{1}{\sqrt{5}}$ and $\tan x < 0$, what is the exact value of $\cos^{-1} x$? Since $\sin x = - \frac{2}{\sqrt{5}}$, we can see that $\tan x$ is in fact $-2$. But how do we ...
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1answer
445 views

Rigorous proof of an infinite product.

I'll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos ...
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1answer
1k views

Finding a result vector from 2 vectors without cross product

If I have 2 lines with its symmetric equations I can get the vectors U and V of each line, and with a cross product I can get the vector R; but how can I get the vector R without a cross product?
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1answer
182 views

Sine rule and equal angles

Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with ...
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0answers
147 views

Solve this trigonometric system $ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $

$$ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $$ I need full solution please. I've tried different transformations, but couldn't get much near, I keep getting huge ...
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1answer
85 views

Elementary trigonometry: $\tan$

How can I assign $a,b,c,d$ values $\pm \tan\theta,\pm{1\over\tan\theta}$ so that ${(a-b)(c-d)\over (a-d)(b-c)}=\tan^2(2\theta)$? Thank you for helping.
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1answer
293 views

Absolute value of $\sin z$ on square

Show that $|\sin z|\geq 1$ at all points on the square with vertices $\pm (N+1/2)\pi\pm(N+1/2)\pi i$, for any positive integer $N$.
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1answer
69k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
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1answer
464 views

Using Pythagorean Identities to Solve for Values

I'm doing homework for my trig class, and it's asking for us to use Pythagorean identities to solve for other trig values. I got through the first 10 fine, but I'm stuck on the last three. My teacher ...
16
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2answers
1k views

De Moivre's Theorem. Motivation and origins.

I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous $$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$ but ...
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0answers
399 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
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1answer
684 views

how to integrate $ \int_{-\infty}^{+\infty} \frac{\sin(x)}{x} \,dx $? [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? How can I do this integration using only calculus? (not laplace transforms or complex ...
16
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3answers
1k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
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4answers
4k views

Solving a triangle given two side lengths and the measure of a non-included angle

Let's say given an angle A = 46 °, side a = 2.29 and b = 2.71 I figured that the angle B = 58.4 by saying: $$B = \sin^{-1} \left(\frac{ 2.71 \sin{46^{\circ}}}{2.29}\right)=58.4^{\circ}$$ But I ...
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1answer
259 views

Finding the direction vectors of two lines that make an angle of 60 degrees

I have this problem, I have to find the values of $a$ so the direction vectors of the lines make an angle of $60$ degrees. $$\frac{x-3}{2} = \frac{y+5}{2} = z+2$$ $$ x-1 = y-1 = ...
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2answers
3k views

Solving a triangle, given two sides and the measure of the included angle

Let say you have a triangle Angle A = 41 degrees , side b = 3.41 and c = 5.83 can you use pythagoras theorem to find the side a? and how can you find Angle B and C
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1answer
424 views

Find angles using the Law of Cosines

if you must find the Angle C based on the sides of a = 2, 3 b = 4,6 og c = 5, 9  I have used the formula: $$\cos (C) =\frac{a^2 + b^2-c^2}{2ab}$$ use, but I think i'm doing something wrong: ...
0
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1answer
1k views

Solving triangles and quadratic equations

When calculating the pieces in a triangle with only two sides and an intermediate angle is known, one must solve a quadratic equation. By solving the equation are 2, 1 or 0 solutions, as ...
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2answers
145 views

Trigonometry & circle math

I tried to solve this Trigonometry question, but I do not know how to solve. I read that the circle has radius 1 and center at (0.0) as the unit circle is plotted in the coordinate system. I ...