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

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1k views

Find a 3D vector given the angles of the axes and a magnitude

I would like to know how one would find a point from the angles of three axes and a magnitude. I know how to do this in 2D: $(\cosΘ * m, \sin(Θ) * m)$. However, I would like to know how this would be ...
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2answers
102 views

MCA entrance question

In triangle $ABC$, the value of $\ \displaystyle \sum_{r=0}^n\ ^nC_ra^rb^{n-r}\cos(rB-(n-r)A)$ is equal to (a) $c^n$ (b) $b^n$ (c) $a^n$ (d) $0$ I have no idea how to start ...
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4answers
101 views

Prove $\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$

How to prove: $\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$ Please help.
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1answer
38 views

$-2(\sin x+2\cos 2x)=0$

I am finding the 2nd derivative critical values for graphing a trig function. So far I have it simplified to $$-2(\sin x+2\cos 2x)=0$$ What values for x make this equal zero? And is there a ...
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0answers
28 views

Trigonometry integration with a bound

So, I want to integrate $\int_\gamma sinz\; dz$ where $\gamma$ is any curve joining $i\to \pi$. Can I say that it is beacause $\int sinz=-cosz$, and $-cosz$ is analytic on the domain containing ...
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1answer
42 views

Angle from known sine,cosine or tangent [closed]

Given the sine, cosine or tangent value, how do I calculate the corresponding angle?
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0answers
26 views

Rotating two objects

I have two lines. Both created in this format: Line 1 $$line1 = \left\{ \begin{array}{c} startX, startY \\ endX, endY \end{array} \right\}$$ $$line2 = \left\{ \begin{array}{c} startX, startY \\ endX, ...
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1answer
107 views

Limit of floor function and sine function

for $$\lim\limits_{x\to k}\lfloor x\rfloor sin\frac{\pi x}2$$ find the limit for $k=0,1,2,3$ i started with $$x-1<\lfloor x\rfloor\le x$$ $$\Downarrow$$ $$xsin\frac{\pi x}2-sin\frac{\pi ...
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1answer
27 views

How to determine the coords for the cell that is on the edge of a grid that intersects a line that extends from a source cell at a given angle?

How do you determine the coordinates for the cell that is on the furthest edge of a grid that intersects an imaginary line that extends from a source cell at a given angle. The grid has a finite size ...
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1answer
41 views

Solve triangle given point and angle

Very basic question, but I'm having difficulty finding an answer. It wasn't listed here, or on $\approx 20$ other sites I looked at. I don't know how to solve a right triangle if I'm only given a ...
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0answers
43 views

I stumbled when i saw this (Travelling Salesman related)

I have here 5 locations like below I then have a tour of 1,2,3,4 (point 5 isn't inserted into tour yet) like below I then find the shortest addition distance to include point 5 into tour, the ...
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3answers
70 views

proving $\lim\limits_{x\to \infty} x\cos\frac1x=\infty$ without using arithmetic

proving $$\lim\limits_{x\to \infty} x\cos\frac1x=\infty$$ and $$\lim\limits_{x\to \infty} x\cos x\neq\infty$$ in $\epsilon,\delta$ form without using arithmetic i am trying to prove that for every ...
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2answers
170 views

Find the area of intersection determined by three circles (Green's Thm)

I'm looking to find the shared area between these three circles using Green's Theorem: $$x^2+y^2=1$$ $$(x-1)^2 + y^2 = 1$$ $$\left(x-\frac{1}{2}\right)^2 + \left(y - \frac{\sqrt{3}}{2}\right)^2 = 1$$ ...
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3answers
845 views

Bearings Problem

I'm presented with the following bearings problem. I believe I have graphed it correctly, although I don't know where to go from here. A US Coast Guard patrol boat leaves Port Cleaveland and ...
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2answers
76 views

Triangle $ABC$ have two sides of length $8$ and $17$. If $\sin 2A=\sin 2B$, find every possible value for the third side.

Ok, I know that the third side has to be smaller than the sum of the other two sides, and larger than the difference of the two sides. But the problem places a limit on the value for the third side. ...
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3answers
72 views

Find the critical numbers of the function.

$$\sin^2(x) + \cos(x)$$ $$\{0 < x < 2\pi\}$$ I thought the answer would be $\pi$, but it is not. Can anyone explain why the answer is not $\pi$?
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2answers
62 views

Trigonometry Direction

Two planes leave an airport at the same time. One plane flies 34° east of north at a speed of 350 miles per hour. The second plane flies 72° west of north at a speed of 275 miles per hour. How far ...
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6answers
155 views

Solutions to $x \sin x=1$ in the interval $0 < x \leq 2\pi$

If I'm in an exam and do not have access to any sort of a calculator, how would I solve it? What method is applicable here or do I have to manually plot points??
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1answer
64 views

what about $\lim\limits_{x\to0}-\frac{\sin x}x=$?

we all know that: $\lim\limits_{x\to0}\frac{\sin x}x=1$ so what is the negative $\lim\limits_{x\to0}-\frac{\sin x}x=$? i am trying to prove what about $\lim\limits_{x\to0}\frac{x^2\sin \frac ...
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1answer
40 views

Interference beats for a more general trigonometric sum

Suppose I have three frequencies $\alpha,\beta,\gamma$ that are all close in value, and I consider the sum $\sin(\alpha x) +\sin(\beta x) +\sin(\gamma x)$ If there were only summands I could find a ...
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2answers
114 views

Domain and range of a function.

Find the domain and range of the function $$f(x)=\frac{1}{\sqrt{[\cos x]-[\sin x]}}$$ Where [] denotes the greatest integer function. I started as $[\cos x]-[\sin x]\gt0$ $\implies \cos ...
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1answer
60 views

Simple vector transformation

I have a question that's probably very simple to a mathematician, but my college days are now far behind me and I'm not sure exactly how to implement this. I'm writing a Java application for the Leap ...
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4answers
162 views

Why is $\int_{0}^{2\pi} |\sin x| dx = 4$

I can't understand why $$\int_{0}^{2\pi} |\sin x| dx = 4$$while $$\int_{0}^{2\pi} \sin x dx = 0$$ I did the calculus for the second varian but I can't reach result $4$ for the first integral. Thank ...
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1answer
47 views

Critical Numbers Problems

Okay so I found the critical number no problem, it being cos x=-1/2, but on my answer sheet it says that the critical numbers are ...
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1answer
88 views

Trigonometric functions over arbitrary angles

Trigonometric functions over obtuse or arbitrary angles doesn't make sense. We can only imagine for eg. sin(x) for angles < 90 degrees because it represents the ratio of the opposite and ...
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1answer
44 views

Trig function evaluations. $\frac{\cos^3 (\pi)}{3}$

I know $\cos (\pi) = -1$. But the $\cos$ to the 3rd power is messing me up. I'm not sure what to do with that. Also, as a note, the entire function $\cos^3 (\pi)$ is divided by $3$.
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0answers
133 views

Integer Factorization via Trigonometry

Nearly 20 years ago, I was sitting in a physics class in high school when a "dumb" question occurred to me: If two pendulums with unknown (different) frequencies started oscillating at the same time ...
2
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4answers
103 views

Trigonometric limit $\lim_{x\to0} \frac{\tan^2{(3x)}+\sin{(11x^2)}}{x\sin{(5x)}}$

How to solve this limit: $$\lim_{x \rightarrow 0}\frac{\tan^2{(3x)}+\sin{(11x^2)}}{x\sin{(5x)}}$$
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3answers
144 views

need to simplify a trigonometric expression

need to simplify this. $$\tan20^{\circ}\cos50^{\circ}+\cos40^{\circ}.$$ I have tried to express $\cos40^{\circ}$ in terms of $\sin20^{\circ}$ and $\cos 20^{\circ}$ but that does not help.
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10answers
11k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
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3answers
44 views

What would be the sum of Trigonometric Functions

I was trying to remember the functions provided on the site: http://www.purplemath.com/modules/idents.htm#restatement From there I came to know about some of the function, basically the Sum of the ...
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2answers
206 views

Solving third degree equation involving trigonometric functions

$2\sin^3 x=\sin x-\cos^2 x+1$. Solve for $x$. I was able to turn it into a quadratic equation, and obtain the answers of $90$, $210$, and $330$ degrees. But the equation has six zeroes.
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6answers
765 views

Verify the identity: $\tan^{-1} x +\tan^{-1} (1/x) = \pi /2$

Verify the identity: $\tan^{-1} x + \tan^{-1} (1/x) = \frac\pi 2, x > 0$ $$\alpha= \tan^{-1} x$$ $$\beta = \tan^{-1} (1/x)$$ $$\tan \alpha = x$$ $$\tan \beta = 1/x$$ $$\tan^{-1}[\tan(\alpha + ...
3
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2answers
66 views

Product of $1-\operatorname{cis}(2k\pi/n)$

I'm in a question about polygonals and got stuck at a part. I have to prove that $$\prod_{k=1}^{n-1} \left(1 - \operatorname{cis}(\frac{2k\pi}{n})\right) = n$$ I've tried to multiply it to make ...
2
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1answer
121 views

Putnam inspired problem

The following is a beautiful problem from Putnam 2003 minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$ I was thinking about a small variation of the above problem minimize $|\sin ...
3
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1answer
363 views

Evaluating trigonometric integral and Cauchy's Theorem

I am trying to evaluate the following integral: $\int_0 ^\pi {d\theta\over{1+\sin^2\theta}}$ I tried using the substitution of $\sin\theta={1\over 2i}(z-1/z)$, where $z=e^{i\theta}$, and ...
0
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1answer
58 views

Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
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3answers
154 views

Evaluating $\int \frac{\operatorname d \! x}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$

How do you integrate $$\frac{1}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$$ or simply $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
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2answers
58 views

Complex and Trigonometric Identities

How can I get this result: $$\frac{1+cis\theta}{1-cis\theta}=-\frac{1}{i\tan(\theta/2)}$$ I've tried to expand $1-cis\theta$ as $(1+cis(\theta/2))(1-cis(\theta/2))$, but it doesn't help.
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1answer
1k views

Taylor Series Expansion for $\tan x$

I'm trying to determine the Taylor series expansion for $\tan x$: I know that the $n$th derivative of the expansion must be the same as the $n$th derivative of the function. Please help, I have no ...
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2answers
60 views

Find the range of arcsin$((1-x^2)^{0.5})$

Title says it all, how do you get the answer to this? So far I only reach $0<1-x^2<pi/2$ but I get an invalid answer from here. the correct answer is $0<x<pi/2$. Any help is appreciated, ...
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1answer
31 views

Relations between trigonometric functions of $a, a/2, a/4, 3a, 6a, 12a$

Problem 1 Given $\sin \left(6a\right)=-\frac{\sqrt{5}}{3}$ and $\cos \left(6a\right)>0$, Find $\sin \left(3a\right)$ and $\tan \left(12a\right)$ Problem 2 Given $\sin ...
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1answer
33 views

Possible trig identity?

Is there a trigonometric identity for $\sin(ab)$? Thanks in advance! I can't find it anywhere. Bothering me a lot. For that matter, what about $\sin(a^{-1})$? Both of these for cosine, too, but if ...
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1answer
22 views

Arc Tangents and Equation

For one of the problems in my book, it requires you to put the arc tangent into the 2piK equation and solve for the arc tangents and lie in [0,2pi]. For: arctan(117)+piK the answers are 1.5622 and ...
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1answer
24 views

FInd the sine and cosine of $2x$ given $\tan x = 3$ and $\sin x < 0$

Find sine / cosine of $2x$, given $\tan x = 3$, $\sin x < 0$ The answer is $\cos 2x = -4/5$ and $\sin 2x = 3/5$ But why is $\cos 2x$ negative? What does $\sin x < 0$ mean?
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1answer
31 views

Find $\sin \left(x\right)$ given $\cos \left(2x\right) $ and an interval for $x$

Find $\sin \left(x\right)$ given $\cos \left(2x\right)=\frac{2}{3}$ and $\pi <x<\frac{3\pi }{2}$ I am trying to solve this problem by drawing a triangle in the appropriate quadrant. ...
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1answer
879 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
8
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2answers
358 views

evaluation of $\int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$

Compute the indefinite integral $$ \int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\,dx $$ My Attempt: First, convert $$ \frac{\cos x+\sin x}{\cos x-\sin x} = ...
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1answer
66 views

How long is the diagonal of this trapezoid?

Given a trapezoid $abcd$, with $|ab| = 1$, and angles $\angle dab = 3\theta/4$, $\angle abc = (\pi + \theta)/2$, $\angle bcd = (\pi - \theta)/2$, and $\angle cda = \pi - 3\theta/4$ (see figure below), ...
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1answer
207 views

Conflict between $\pi$ and ($\sqrt2/81) \times 180$

Conflict between ${\pi}$and ($\frac{\sqrt2}{81})\times 180$. $\frac{\sqrt0.5}{40.5}$ = $\frac{\sqrt2}{81}$. If I have a number 486 per example and I divide 486 by 40.5 and then by $\sqrt2$ ,I would ...