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

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662 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
73 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
66 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
46 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
56 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
32 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
101 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
58 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
153 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
44 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
69 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
42 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
108 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 ...
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4answers
93 views

Trigonometric Limit with $\sin{11x^2}$

How to solve this limit: $$\lim_{x \rightarrow 0}\frac{\tan^2{(3x)}+\sin{(11x^2)}}{x\sin{(5x)}}$$
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3answers
140 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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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
39 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
169 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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480 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 + ...
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2answers
60 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 ...
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1answer
109 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 ...
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1answer
268 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 ...
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1answer
56 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
139 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
54 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
747 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
58 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
30 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
32 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
21 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
28 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
579 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 ...
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2answers
312 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
65 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
173 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 ...
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1answer
46 views

Calculate regular or equilateral triangle altitude with radius only possible?

I need to calculate the altitude of a regular triangle (equilateral) but i only have the radius (polygon radius) available (http://www.mathopenref.com/polygonradius.html). I have been searching for ...
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2answers
110 views

Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$

Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$. I tried some elementary ways while all failed.
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1answer
38 views

Another Verify the identity: $\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$

Another Verify the identity that I can't get: $$\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$$ $$ = \frac{1 + \left(\frac{1}{\cos x}\right)}{2}$$ $$ = \frac{\cos x + 1}{2 \cos x}$$
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4answers
160 views

Verify the identity $\sin 3x + \sin x = 4\sin x - 4\sin^3 x$

Verify the identity: $$\sin 3x + \sin x = 4\sin x - 4\sin^3 x$$ All I've done is this, and I don't know where to go from here: $$4\sin x - 4\sin^3 x$$ $$ = 4(\sin x - \sin^3 x)$$ $$ = (4\sin x)(1 ...
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1answer
40 views

Verify the identity: $2\cos^4 x - \cos^2 x - 2 \sin^2 x\cos^2 x + \sin^2 x = \cos^2 2x$

Verify the identity: $$2\cos^4 x - \cos^2 x - 2 \sin^2 x\cos^2 x + \sin^2 x = \cos^2 2x$$ There are so many routes to start on this. I have tried a bunch, have gotten stumped at each one.
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3answers
336 views

Verify identity $\sin 2x - \cot x = -\cot x \cos 2x$

Verify the identity: $\sin 2x - \cot x = - \cot x \cos 2x$ I haven't gotten very far: $2 \sin x \cos x - \frac{\cos x}{\sin x}$ ...
3
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2answers
398 views

Find intersection of line segment in rectangle perimeter.

I want to find the best way to calculate the point in the perimeter of a rectangle in which a line segment intersects. p is a point inside the rectangle ($(0, 0)$ is in the center of the rectangle). ...
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2answers
81 views

$3\cos(x+1) =\cos(x+2)$. This is a equation, involving trigonometric functions.

This actually derives from the same equation though without the parentheses. Honestly, I haven't learn about the stuff yet. And I don't know if there is a answer for the problem.
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68 views

Trigonometry question (given that cosec x = 9)

Given that cosec x = 9 without using a calculator, evaluate (a) cot x (b) tan x, and (c) cos x
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1answer
96 views

Trigonometry question finding the sin and tan

If angle $\alpha$ is reflex, and $\cos \alpha = -\frac{9}{41}$, without using a calculator, evaluate (a) $\sin \alpha$ (b) $\tan \alpha$ (c) $\cos (\alpha - 180\deg)$ .
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1answer
46 views

Sin x tends to x as x tends to zero

If ABC is a sector of a circle with centre A, why does the area of triangle ABC approach the area of sector ABC as angle BAC approaches zero?
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1answer
70 views

Limiting Function: Composition of Sine Curves

Define $f_1(x) = \sin(x)$. For each $k \geq 1$, define $f_{k+1}(x) = (f_k \circ f_1)(x)$. Are the properties of the limiting function (if it exists) well-known? That is, as $n$ tends to infinity, ...
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2answers
33 views

I need help to see if the corresponding angles are correct

$\displaystyle\cos 4x - \cos 3x=0$ $\displaystyle-2 \sin (7/2)x \cos (1/2)x= 0$ So does $\displaystyle\sin (7/2)x=\dfrac{2\pi}{7}$ and $\displaystyle\cos (1/2)x=\dfrac{\pi}{2}$?