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

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3
votes
2answers
186 views

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
0
votes
3answers
987 views

Distance between two antennas

I am trying to find out the formula to calculate how high antennas need to be for Line of Sight (LoS) propagation. I found: d = 3.57sqrt(h) also ...
7
votes
6answers
439 views

Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $

My attempt: $\int {{{1 + \sin x} \over {\cos x}}dx} $, given : $u = \sin x$ I use the general rule: $\eqalign{ & \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr ...
6
votes
2answers
79 views

Another trigonometric proof…?

...sigh..another problem how shall I prove the following? $$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$ so what now? the following's what I've done: $$\cot A - \cot^2 A ...
1
vote
6answers
75 views

How to prove this trigonometric expression?

How would you go about proving the following? $${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$ This is what I've done so far: $$LHS = {1+\cos^2 A -2\cos A + 1 - ...
-1
votes
2answers
56 views

For a complex number $z$ applies to $\operatorname{Re} (z) = 5$.

For a complex number $z$ applies to $\operatorname{Re} (z) = 5$. What values can $\operatorname{Re} (1/z)$ assume to be?
2
votes
2answers
66 views

Prove that a sum converges to a trigonometric expression

$$2^n \cos \left (\frac{n \pi}{2} \right )=\sum_{k=0}^{n} (-1)^k \binom{2n}{2k}$$ I expanded the LHS and got $$\binom{2n}{0}-\binom{2n}{2}+\binom{2n}{4}-\binom{2n}{6}+\cdots+(-1)^{n}\binom{2n}{2n}$$ ...
1
vote
1answer
1k views

Phase Relationships of Sinsuoidal Waveforms?!

I'm reading through a chapter on sinusoidal alternating waveforms and I'm having some difficulty in the section on Phase Relations. The generic expression is mentioned below for a waveform that has ...
14
votes
3answers
269 views

Proving a trig infinite sum using integration

How can I prove the following using integration and elementary functions? Prove that: $$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$ $0 < \theta < 2\pi$
5
votes
1answer
145 views

Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
0
votes
1answer
4k views

Calculate new positon of rectangle corners based on angle.

I am trying to make a re-sizable touch view with rotation in android. I re-size rectangle successfully. You can find code here It has 4 corners. You can re-size that rectangle by dragging one of ...
8
votes
4answers
270 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
2
votes
6answers
140 views

Solve the following limit?

Please only give hints? $$\lim_{x \to \frac{\pi}{6}}\frac{2\sin{(x)}-1}{\sqrt{3}\tan{(x)}-1}$$ I tried this and I was able to simplify it down to $$\lim_{x \to ...
2
votes
2answers
279 views

Prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$?

I need to prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$. I have already checked that this is correct for $n=1$ and $n=2$, but I'm not able to prove this identity by induction. Now I ...
1
vote
3answers
2k views

Please help me find a formula to find the 3rd point in a right triangle

I'm trying to figure out how to plot a 3rd point on a graph Given the following line segments and angles Is there a formula for the 3rd point? Note: This image is just for an example. The base ...
1
vote
1answer
56 views

Please help me to find an equation to find the 3rd point in an arc.

Long story short, I want to animate the rotation of an object that's based off a circle. Given the center point of the circle, the radius, and one of the points in the arc, is it possible to find the ...
2
votes
3answers
64 views

How can I find the limit of $\lim _{x\rightarrow \infty }\left( 4x^{2}\sin ^{2}\left( \dfrac {2} {x}\right) \right)$ to infinity?

The title basically tells everything. The result is 16 but I can't figure out how to do this. Thanks!
0
votes
2answers
572 views

Whats the maximum value of $ y=6\cos\left( \frac {2\pi}{14} x\right)-2?$

Please show the correct way how to do this thanks. I got this.. $y$ will be maximum when $\cos 2\pi/14x - 2$ is maximum i. e. when $2\pi/14x- 2 = 0$ [ $\cos 0$ is maximum $= 1$ ] so or ...
2
votes
3answers
184 views

Limit without L'Hopital's rule

Please solve this withouth L'Hopital's rule? $$\lim_{x\rightarrow\sqrt{3}} \frac{\tan^{-1} x - \frac{\pi}{3}}{x-\sqrt{3}}$$ All I figured out how to do is to rewrite this as $$\frac{\tan^{-1} x - ...
2
votes
2answers
122 views

Prove this double angle identity?

How would I prove the following double angle identity? $$\frac{\sin2A - \sin A}{\cos2A + \sin A}=\tan \frac{3A}{2} . \cot\frac{A}{2}$$ Sadly I am stuck.
2
votes
6answers
346 views

A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$?

What is the range of the function $$\frac{\sqrt{3}\sin x}{2+\cos x}$$
0
votes
1answer
61 views

Is my technique valid? [duplicate]

I have serious doubts about this, but I thought you guys might at least fix this and suggest something useful which would make this approach work. The question is to prove that $$2=2\cos(x)+x\sin(x)$$ ...
0
votes
1answer
93 views

Prove this proprety of $f(x)$

I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$? Let ...
1
vote
0answers
93 views

Strange results for solving boundary angle of full reflection

I was trying to solve the following group of equations for $\alpha$ using Wolfram|Alpha: $$\frac{v_2\cdot\sin(\alpha)}{v_1} = 1 \\ v_1 > 0 \\ v_2 > 0$$ I expected something like $\alpha = ...
3
votes
4answers
110 views

Trigonometric Formula

I am stuck with the simple expression $$ \frac{\cos^2(\theta + \alpha)}{1 - \cos^2(\theta - \alpha)} = \text{const.} $$ where $\theta$ is a variable and $\alpha$ is the number satisfying $$ \alpha = ...
18
votes
4answers
238 views

Solve for $x$ a trigonometric equation

I want to solve for $x$ $$ {{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x $$ but I don't know how to start. Replacing $\sin x$ or $\cos x$ by $y$ led me nowhere ...
1
vote
0answers
62 views

How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
0
votes
2answers
130 views

Number of real roots of an equation [duplicate]

What is the number of real roots of the equation $$2\cos(\frac{x^2 + x}{6}) = 2^x + 2^{-x}$$ How to solve this kind of problems. Any general methods ??
2
votes
1answer
97 views

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
3
votes
1answer
64 views

Trig and algebra problem: Finding sides of a triangle

Let $ABC$ be a triangle such that $\angle ACB = \pi/6$ and let $a,b,c$ denote the lengths of the sides opposite to $A,B,C$, respectively. What are the value(s) of x for which $a = x^2 + x + 1, b = ...
0
votes
1answer
52 views

Identifying degrees and radians

I have the following problem : If $\sec(1.4) = x$, find the value of $\csc(2\tan^{-1}x)$. (A) $0.33$ (B) $0.87$ (C) $1.00$ (D) $1.06$ (E) $3.03$ I we take the $1.4$ as degrees, we get option ...
0
votes
2answers
60 views

Given three plots of trigonometric functions, find an expression for each function [closed]

Considering the three graphs in figure 1 showing trigonometric functions, $f(x)$, $g(x)$ and $h(x)$. Using these graphs, write the expression for each function.
6
votes
1answer
155 views

Solve $\sin(x)+2\sin(x)\cos(x)=\pi/4$

Is it possible to solve (not approximate) the following trigonometric equation by hand? $$\sin(x)+2\sin(x)\cos(x)=\pi/4.$$
2
votes
1answer
431 views

Cone shaped related rates of change question

A container is in the shape of a cone of semi-vertical angle $30^\circ $, with it's vertex downwards. Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$ At the ...
5
votes
4answers
2k views

Learning trigonometry on my own.

I have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch. I am seeking both books and perhaps lectures on trigonometry and ...
0
votes
1answer
154 views

Solving a trigonometric inequality

I am solving this inequality for $\theta$ $$\cos \theta > -\frac{\rho}{\sqrt{1-\rho^2}} \sin \theta$$ with $\rho \in (-1,1)$ given, when trying to integrate a function under polar coordinates and ...
0
votes
2answers
70 views

the period of a trigonometric function

I'm trying to solve a differential equation which is : $$y'(t)-4y(t) = \cos(3t)$$ Resolution of the equation without the second membre $y'(t)-4y(t)=0$ has as solution $ y_s(t)=ke^{4t} $ with ...
2
votes
5answers
4k views

The hypotenuse of a right angle triangle measures 12 cm. What size angles would produce maximum…

The hypotenuse of a right angle triangle measures 12 cm. What size angles would produce the maximum perimeter? I got to point where I take the derivative and get $12(\cos\theta-\sin\theta)=0$, not ...
0
votes
0answers
131 views

Getting the angle between three points

So I have this psuedo code here (converted from c# to show you better) ...
4
votes
1answer
90 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
0
votes
2answers
192 views

Real and Imaginary Parts of $\frac{\cos(z)}{(1-e^{ix})}$

Find $$\mathrm{Re}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ and $$\mathrm{Im}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ Please help I've been trying for some time now...
1
vote
1answer
79 views

finding Length of a diagonal

Given Quadrilateral ABCD in such that $AB<BC<CD$ creating increasing arithmetic progression with sum of $27$ cm. $\measuredangle BCD=60^{0}$. the diagonal $BD=\sqrt{133}$ cm, and it divided ...
0
votes
3answers
3k views

Calculating circle radius from two points and arc length

For a simulation I want to convert between different kind of set point profiles with one being set points based on steering angles and one being based on circle radius. I have 2 way points the ...
1
vote
2answers
184 views

What is the best way to solve equations with trig functions

I usually use guessing to solve equations with trig functions. Yesterday, I came across an equation that I couldn't really write it in a helpful form to guess. My question is, how can I solve equation ...
1
vote
1answer
53 views

Angle consistency between vectors in N dimensions

I am trying to understand how rotations work in higher dimensions. Let us assume we have a set of points $p_i\in P$ in $N$ dimensions, related to another set of points $q_i \in Q$ by a rotation $R$. ...
2
votes
5answers
906 views

Show that $\sin^{-1}( x) =\tan^{-1}(x/\sqrt{1-x^2})$ for $|x| <1$

Show that $\sin^{-1}( x) = \tan^{-1}(x/\sqrt{1-x^2})$ for $|x| <1$ I have got as far as knowing that values of $\sin^{-1} x$ are only defined when x lies in the set [-1, 1]. and that for any ...
4
votes
1answer
135 views

eye vision problem

Imagine that the smallest letter that ken can read on the Snellen Eye chart is 3 inches tall. What is Ken's vision, using 20/XX notation? I have a question about eye vision, but I have no idea how ...
2
votes
3answers
117 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
3
votes
2answers
234 views

How to find the solutions $x$ of $ 2\sin{11^{\circ}}\sin{71^{\circ}}\sin{(x^{\circ}+30^{\circ})}=\sin{2013^{\circ}}\sin{210^{\circ}}$

Let $$2\sin{11^{\circ}}\sin{71^{\circ}}\sin{(x^{\circ}+30^{\circ})}=\sin{2013^{\circ}}\sin{210^{\circ}}$$ where $90^{\circ}<x<180^{\circ}$. My idea: ...
0
votes
1answer
21 views

What is Angle(A,b) about something.

I was reading a paper and came through a notation saying .... Angle = Angle(A,B) about C. Can anybody tell me what exactly it means. Thnaks, Harsha