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

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9
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2answers
218 views

How to Solve Trigonometric Equations?

How are you supposed to go about solving equations such as: $$-\sqrt{3} = \frac{\sin{4\theta}}{\sin{7\theta}}.$$ I know that $\theta = 30^{\circ}$ is one such solution, but how do I find all ...
5
votes
3answers
64 views

Range of trigonometric functions

I would like to know if there is a simple approach to find the range of functions in the form: $$\sin x\sin2x$$ $$\cos x\cos3x$$ $$\sin 2x\cos 4x$$ For example, finding the range of a function in ...
0
votes
1answer
9 views

Find Adjacent only knowing Angle and Opposite

Can you find the length of the adjacent side of a right triangle only knowing the length of the opposite side and the angle? If so how do you calculate it?
1
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4answers
35 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
3
votes
1answer
50 views

Triple Angle Condition

Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter. Essentially we want sinb, sin3b, sin4b to have rational ratios (manipulate ...
1
vote
1answer
26 views

Data transformation of angles such that $90^\circ$ is equal to $-90^\circ$

Is there a transformation I can perform on a dataset of angles (from $-90^\circ$ to $90^\circ$) such that the transformation of $-90^\circ$ is equal to that of $90^\circ$? I am only interested in what ...
1
vote
2answers
21 views

Centripetal acceleration for a polyline

Given a polyline (x and y coordinates) in Cartesian coordinate system and time component, how can I estimate centripetal acceleration (let's say an average one)? (I have a list of pairs. Each pair ...
0
votes
1answer
21 views

How to find the domain of this trig function?

f(x)=sqrt(tan(2x+π)) Allright, so i know you cannot have a number less than zero under the square root sign and that tangent cannot equal π/2+nπ. So should i try to find the domain of the tan ...
2
votes
2answers
33 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
-1
votes
1answer
27 views

How to show the following inverse trigonometric equation? [closed]

Let $a$ and $b$ be real numbers so that $ b\neq 0 $. Show that $ \dfrac{tan^{-1}\left( \dfrac{a}{b}\right)}{\pi}=\dfrac{\ln \left( b/ \sqrt{a^{2}+b^{2}}+ai/ \sqrt{a^{2}+b^{2}}\right) }{\ln (-1)}$.
1
vote
3answers
38 views

Translate a point on a circumference

If I have a point $A$ on the circumference of a circle with origin $O$ and radius $r$, how would I find the coordinates of point $B$, which is also on that circumference, but is $d$ units away from ...
0
votes
3answers
38 views

Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$

Let $z=\cos\theta+i\sin\theta$. Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$ Can anyone show me how to show the equation? I can't think of how to get $\frac 12 ...
0
votes
0answers
16 views

An isosceles triangle of wood is placed in vertical plane, vertex upwards and faces Sun. If 2a be base of triangle…

Problem : An isosceles triangle of wood is placed in vertical plane, vertex upwards and faces Sun. If 2a be base of triangle, h its height and $30^{\circ}$ altitude of Sun, then prove that tangent ...
1
vote
0answers
12 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 ...
16
votes
4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
0
votes
1answer
18 views

Trig Word Problem Involving System of Equations

Hobbyists often compete with their model rockets to determine which rocket flies the highest. On one test launch, a rocket was fired vertically upward. The angle of elevation to the top of the flight ...
0
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0answers
28 views

Triangles congruency exercise

This exercise tells that: $$AD = AE, \angle A \cong \angle DEC, \angle ADE \cong \angle BDC$$ Then I have to show that $$\triangle ADB = \triangle EDC$$ The exercise solution highlights that we ...
0
votes
2answers
24 views

Are there infinitely many pairs $(x, y)$ satisfying $\cos(x+y)=-1$?

Consider the equation $$\cos(x+y)=-1.$$ I think $\cos(x+y)=-1$ only when $x+y=(2n+1)\pi$ for $n$ any integer. If $S$ is the set of all pairs $(x, y)$ in $\mathbb{R}^2$ satisfying $\cos(x+y)=-1$, is ...
0
votes
1answer
37 views

If the cos of 27 is 0.89, how much is the csc of 27

Hey guys for my trig class we're viewing trigonometric functions and their properties. So far I have understood but I came across this problem and can't seem to solve it: Given the approximation cos ...
0
votes
3answers
50 views

Trig Substitution Integral Question

My class is going over trig substitution, but I can't figure this one out, mostly because it's not in the correct form. Could someone help explain how to set up this problem? $$ \int \frac ...
9
votes
7answers
140 views

Finding $\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$

How would I go about solving this following limit? $$\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$$ My attempts: Direct substitution yields the limit to be undefined, also ruling out the ...
0
votes
0answers
36 views

Trigonometry graph sketch [closed]

Sketch the graph of $y=|\tan x|$ for $0 \leq x \leq 360^{0}$ showing clearly the position of the asymptotes. Solve the inequality $|\tan x| < 1$ for $0 \leq x \leq 360^{0}$.
2
votes
4answers
39 views

Use substitution to solve for $x$ in $\frac{1}{2-\sin x}=\sin x$

Use substitution to solve for $x$ in the following equation: $$\frac{1}{2-\sin x}=\sin x$$ This is what I have done so far: $$\sin^2x-2\sin x+1=0$$ $$\arcsin(1)=\frac{\pi}{2}=x$$ The correct ...
1
vote
2answers
39 views

Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $

How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for $0< A< \pi /4$? This is one of the problems in a book I'm using. It is ...
0
votes
2answers
34 views

Finding $|a|$, a complex number, given a system of equations

$a$ and $b$ are complex numbers where $|2a - b| = 25$, $|a + 2b| = 5$, and $|a + b| = 2$. Using the information, find $|a|$. I tried using the magnitude formula (i.e. where $|a| = \sqrt{x^2+y^2}$), ...
0
votes
1answer
34 views

How to solve equations like $x/5 + \cos2x = 2$?

I'm going through an old test paper. Some more info was given: the answer lies in the interval $-20\le x\le 20$. There are several solutions, the question is to find the smallest and the total number ...
0
votes
2answers
11 views

Finding the set of points of a polar coordinate

$\left\{ (r,\theta) : 2\le r\le 6,\frac{\pi}{3}\le\theta\le\frac{5\pi}{6}\right\}$, where $S$ stands for the set of points. What is the area of $S$? This is a bit confusing to me. How do I start ...
2
votes
2answers
23 views

Converting (7,5) Cartesian coordinates to polar coordinates

Find the point (r, $\theta$) in polar coordinates given the fact that when converted in Cartesian coordinates, the point is $(7,5)$. Use that to find the point $\left( 2r, \theta + \frac{\pi}{2} ...
0
votes
1answer
18 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
4
votes
3answers
49 views

Trigonometric functions - finding solutions

Question: Find the general solution for the equation: $$\sin x + 2\sin2x - \sin3x = 3$$ Approach: Well using the identity of $\sin a - \sin b $, I merged together $\sin x - \sin3x$ And as ...
1
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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?
0
votes
1answer
26 views

Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D

this is my first post.. I hope this good I have 1 triangle in space (3D)... and I know all data except the coordinates of 3er point(vertex)... for example this: then: ...
0
votes
0answers
25 views

Given $4$ points in the space, how do you check if an arbitrary point is within the area marked by those points?

Given $4$ arbitrary points in the space $A(x_1,y_1), B(x_2,y_2), C(x_3,y_3,), D(x_4,y_4)$, how do you check if an arbitrary point $X(x_5,y_5)$, is within the quadratic area marked by the $4$ points ...
-1
votes
1answer
21 views

Find the domain of this function [closed]

How would I go about finding the domain of this function? $$f(x)= \sqrt{\tan(2x+\pi)}.$$
0
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0answers
73 views

How to solve for $t$ using $\arctan(t)$

Context: I solved the following simple nonhomogeneous equation: $y''-3y'-4y=2 \sin (t)$. We assumed that $Y(t)=A\sin(t)$, where $A$ was to be determined. Now, $Y'(t)=A\cos(t)$ and ...
1
vote
2answers
54 views

Proof that a Limit equals a Function

Let f(x) = $\frac{x}{|x|}$ if x $\neq$ 0, and define f(0)=0. Show that f(x) = $\lim_{n \rightarrow \infty} \frac{2}{\pi} \tan^{-1}(nx)$ My work: $\frac{x}{|x|}$ = $\lim_{n \rightarrow \infty} ...
0
votes
0answers
41 views

Finding $\lim\limits_{(x,y,z) \to (1,2,-3)} \arctan\left(\frac{x+z}{y}\right)$

this is a homework problem, so I am just looking for a hint to get me going in the right direction. I am asked to find the following limit and prove my result, or to show that the limit does not ...
2
votes
4answers
327 views

Does this limit not exist?

I used u substitution for the $\lim_{x \to\infty}x^2 \sin\frac{1}{x}$ and got the limit does not exist by saying $u=\frac{1}{x}$. Is this correct and if so would that mean $\lim_{x \to\infty}x^3 ...
0
votes
2answers
70 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
0
votes
0answers
8 views

Organizing / clustering nodes in 3d space when angle of direction is given

The Problem I am working on a problem of clustering for mobile nodes. Initially, I though of clustering the nodes based on their angle of movement (relative to the x-axis). However this situation ...
3
votes
3answers
148 views

What is the limit of $x/(x+\sin x)$ as $x$ approaches infinity?

I am trying to determine $$\lim_{x \to \infty} \frac{x}{x+ \sin x} $$ I can't use here the remarkable limit (I don't know if I translated that correctly) $ \lim (\sin x)/x=1$ because $x$ approaches ...
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votes
2answers
35 views

Here is a exam paper question unit 3 exedel [closed]

Here is the picture Could i have a step by step explanation of how to work it out? Thanks
7
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5answers
88 views

$\sec\theta$ never equals $\tan\theta$. Or does it?!?

Wolfram-alpha is disagreeing with me. I am wondering why: which of us is wrong? I was writing a question to demonstrate why domains of functions matter. The question is: Does there exist an angle ...
0
votes
2answers
42 views

Calculate the length of the adjacent side when $\cos \angle{BAC} = \frac{3\sqrt{109}}{109}$

I am learning basic trigonometry. I've been doing this exercise: And I am told that $$\cos \angle {BAC} = \frac{3\sqrt{109}}{109}$$ Well, the $?$ is the adjacent side of $\angle{BAC}$, so I am ...
1
vote
3answers
55 views

Invert a $2\times 2$ Matrix containing trig functions [duplicate]

Invert the $2\times 2$ matrix: \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} My thought was to append the $2\times 2$ identity matrix to the right ...
0
votes
3answers
34 views

A problem in calculus mean value theorem

Hi tried to solve this for hours, any idea how to approach this question: prove for every $x>0$ $$2x\times\arctan(x)>\ln(1+x^2)$$
2
votes
2answers
41 views

What is the equation for this wave?

So it would be hard to describe it, it's better to see it yourself: http://physics.info/waves/surface-wave.html (Angular velocity of rotating points is constant I presume) What is it called? What ...
2
votes
2answers
89 views

How do you solve the equation $ (z^2-1)^2 = 4 ? $

$ (z^2-1)^2 = 4 \iff $$z_1 = 3 $ and $ z_2=-1$ $arg(z_1)= 0 , arg(z_2) = \pi$ $$ z_1 = \sqrt{3} \left(\cos\left(\frac{2\pi k}{2}\right) + i\sin\left(\frac{2\pi k}{2}\right)\right)$$ $$z_2 = i ...
2
votes
1answer
18 views

Triangle Inequality with Vectors

If the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ are $5$ and $12$, respectively, then the magnitude of vector $(\mathbf{b-a})$ could NOT be (A) 5 (B) 7 (C) 10 (D) 12 (E) 17 The triangle ...
1
vote
3answers
58 views

Problem Involving $2$ Right Triangles Trigonometry

Can't really figure out this problem where I have to find side $RS$, this is covered in my high-school trig curriculum, and it is in the section which deals with all concepts before sine and cosine ...