4
votes
1answer
27 views
Polar coordinations - problem with r and $\theta$
let's take a look on Archimedean spiral.
the polar equation is $r = \theta$. click here to look.
but $\tan (\theta) = y/x$ and $r = \sqrt{x^2+y^2}$,
so $r = \theta \rightarrow \tan(\sqrt{x^2+y^2}) ...
3
votes
0answers
37 views
looking for reference or nice proof of trig lemma
Math people:
I am looking for a reference or a nice proof of the following fact. I have proven it myself, but my proof is messy: let $\theta \in (0,1]$ and $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
4
votes
1answer
61 views
How can I prove that $\sin (10^\circ), \sin(1^\circ), \sin(2^\circ), \sin(3^\circ), \tan(10^\circ)$ are irrational
How can I prove that $\sin (10^\circ), \sin(1^\circ), \sin(2^\circ), \sin(3^\circ), \tan(10^\circ)$ are irrational?
My try:: Let $x = 10^\circ$, Then $3x = 30^\circ$
Now $\sin (3x) = \sin ...
2
votes
2answers
37 views
Trigonometric Identity Problem - Cos Tan and Sin
I have been going through my lecture notes for a structures question (the solution of a 2nd order ode for a buckling problem) when I came across a very weird trigonometric simplification which I just ...
4
votes
1answer
38 views
How to find area of triangle from its medians
The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is
a) $48$
b) $144$
c) $24$
d) $72$
I don't want whole solution just give me the hint how ...
1
vote
2answers
43 views
Trigonometric problem
I'm trying to get the roots for a complex number $x^2+1$
$x^2+1=0\rightarrow x^2=-1 \rightarrow x = \sqrt{-1} \rightarrow i$
So, $w^2 = 0 + 1i$
$p = \sqrt{0^2+1^2} = 1$
$\theta = \tan^{-1} \left( ...
0
votes
2answers
28 views
Trigonometrical Question
the question is solve the following equation in the interval
$$0<\theta\leq 360$$
$$\tan(\theta) = \tan(\theta)(2+3\sin(\theta))$$
I got 199.5 and 340.5 as my answers like so:
$\tan(\theta) = ...
4
votes
2answers
68 views
How does a calculator calculate the sine, cosine ,tangent using just a number?
Sine Θ = oposite/hypotenuse
Cosine Θ = adjacent/hypotenuse
Tangent Θ = oposite/adjacent
So in order to calculate the Sine or the cosine or the tangent I need to ...
2
votes
3answers
27 views
Right-angled isosceles triangles
If a right-angled triangle is isosceles then the other two angles must be equal to $45^\circ$ ?
Is this always the case or are there other possible right-angled isosceles triangles?
1
vote
1answer
36 views
Find next point in ellipse given the chord length
I would like to draw a cloud programmatically. For this reason I need to know where to draw the next circle around the ellipse.
Given the chord (circle radius), how can I calculate the next point in ...
0
votes
1answer
43 views
Math word problem. Any help is appreciated.
A math student writes a proof of the derivative of a certain trigonometric function. The last line she writes before stating her conclusion is...
$\dfrac{d}{d\theta} \left(\sin\left(\theta ...
3
votes
3answers
46 views
Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx} $
$\eqalign{
& \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr
& = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr
& = {\int {\left( ...
1
vote
1answer
72 views
$\pi$ is just a number, or also the circumference of a sub-unit circle?
A unit circle defined in the Cartesian plane has a radius of $1$ and a diameter of $2$. So making a full round is $2 \pi$. Now, $\pi$ is the ratio of the circumference over the diameter, so if I have ...
5
votes
4answers
70 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
1
vote
3answers
81 views
Proof using trigonometry that circle circumference is $2 \pi R$
Using trigonometry, I would like to prove that the circumference of a circle is $2\pi$ times its radius. Can someone help please?
6
votes
3answers
103 views
How can I find all the solutions of $\sin^5x+\cos^3x=1$
Find all the solutions of $$\sin^5x+\cos^3x=1$$
Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
7
votes
2answers
48 views
How to show this inequality?
Show that $$-2 \le \cos \theta(\sin \theta+\sqrt{\sin^2 \theta +3})\le2$$
Trial: I know that $-\dfrac 1 2 \le \cos \theta\cdot\sin \theta \le \dfrac 1 2$ and $\sqrt 3\le\sqrt{\sin^2 \theta ...
2
votes
1answer
29 views
Trigonometry Word Problem--Not sure if correct
From point $A$ the angle of elevation to the top of a newly constructed building is $17.2$ deg. From point $B$ which is $153$ meters closer to the building the angle of elevation at the top of the ...
0
votes
3answers
37 views
Force required to push an object?
What is the force required to push a 1000 lb object up a ramp that is inclined at a 40 degree angle?
0
votes
1answer
60 views
Why do these trig functions “overpower” each other?
For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
1
vote
1answer
20 views
Circular motion trig
We have $x_P = -2 + 4 \cos (-\pi t)$ and $y_P = 1 + 4 \sin ( - \pi t)$ with $t$ in seconds.
We have to find the coordinates of the intersection with the y-axis. So I use trig and I eventually end up ...
1
vote
2answers
33 views
Can you find the resultant force between these two vectors?
Determine the magnitude of the resultant force on an object if force $A$ is pulling the object with $150$ lbs of force and force $B$ is pulling with $300$ lbs, and the angle between the two forces is ...
1
vote
1answer
32 views
Find the value of $\tan^2\alpha+\cot^2\beta$
A circle with centre o have two chords AC and BD,which are intersecting each other at P.If $\angle AOB=15^\circ$ and $\angle APB=30^\circ$,then find out value of
$$\tan^2\angle APB+\cot^2\angle COD$$
...
0
votes
1answer
17 views
Find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550)
I am attempting to find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550).
Is my calculation correct because a program is giving me a different answer? It says the angle is ...
-1
votes
4answers
46 views
Find the value of $\frac{\tan\theta}{1-\cot\theta}+\frac{\cot\theta}{1-\tan\theta}$ [duplicate]
I want to know an objective approach to solve these type of expression in a quick time
Which of the expression equals to
$$\dfrac{\tan\theta}{1-\cot\theta}+\dfrac{\cot\theta}{1-\tan\theta}$$
...
2
votes
1answer
23 views
Simplify difference of two arc tangents?
I have a problem, that I am trying to simplify, but there does not seem to be something obvious regarding it.
Very simply, I am trying to figure out if there is a way to 'open' the following:
$$
...
1
vote
0answers
10 views
Can you help me reverse the Minimum Curvature Method?
The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
2answers
31 views
Integral of $\int \frac{\sin(x)dx}{3-\cos(x)}$
I am trying to solve this integral and I need your suggestions.
I don't know if its OK to set $3-\cos(x)$ as $t$ $\rightarrow dt = \sin(x)dx$ or just take $\cos(x)$ and set it as $t$
$$\int ...
2
votes
4answers
61 views
Definite integration of a trigonometric function
How to integrate $$\int_0^{\pi/2}\!\dfrac{2a \sin^2 x}{a^2
\sin^2 x +b^2 \cos^2 x}\,dx $$
my first step is $$\frac{2}{a} \int_0^{\pi/2}\!\dfrac{a^2 \sin^2 x}{a^2 +(b^2 - a^2) \cos^2 x}\, dx $$
I ...
11
votes
3answers
144 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
1
vote
1answer
15 views
Find the equation of the hyperbola given foci and the minor axis
first time posting and using the site. I have a quick problem that I need some help with. I need to find the equation of a hyperbola given the foci and the length of the minor axis.
The foci ...
10
votes
5answers
348 views
What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
2
votes
1answer
40 views
A trigonometric identity for special angles
Prove that for a natural number $n$,
$$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
3
votes
2answers
64 views
Why is this derivative incorrect?
We have to find the derivative of $$f(x) = \dfrac{\tan(2x)}{\sin(x)}$$
I would like to know why my approach is incorrect:
$$f'(x) = \dfrac{\sin(x) \cdot \dfrac{2}{\cos^2(2x)} - \tan(2x) \cdot ...
6
votes
2answers
37 views
relationship of polar unit vectors to rectangular
I'm looking at p. 16 of Fleisch's Student's Guide to Vectors & Tensors. He's talking about the relationship between the unit vector in 2D rectangular vs. polar coordinate systems. He gives these ...
2
votes
3answers
39 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
0
votes
0answers
54 views
Solution procedure for a system of trigonometric equations in two variables
i would like to know if there's a method for solving the following system using (or not) tan half angle substitution.
$$A\cdot\sin(\theta_1) + B\cdot\cos(\theta_1) + C\cdot\sin(\theta_3) + ...
3
votes
3answers
101 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
4
votes
2answers
83 views
Help me solve a trigonometric equation
I am doing some work in RF circuit design. I need to solve an equation for my design:
$$\frac 1{\cos(t_1)}+\frac 1{\sin(t_1)} =\frac 1{\cos(t_2)}+\frac 1{\sin(t_2)}$$
(I created a nicely typed image ...
1
vote
1answer
25 views
Trigonometry Addition Thereom With Only one exact value?
Use the expression of $\sin(A+B)$ to evaluate $\sin 195$.
Do I use one exact value like $45+150$ or $60$ or is there another way?
0
votes
2answers
23 views
Trigonometry Addition Thereom
Using the expansion of
a. $\sin(𝐴+𝐵)$, prove that $\sin75°=\sqrt 6+\sqrt{24}$
b. $\sin(𝐴+𝐵)$, prove that $\tan75°=2+\sqrt 3$
Where to start? draw up triangle of sin 75? find other values? help ...
1
vote
1answer
63 views
Trigonometric equality $x = 99 \sin (\pi x)$
Find the number of real solutions of $\displaystyle x = 99 \sin (\pi x)$.
I am getting stuck in some trigonometric relations.
8
votes
1answer
83 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
15
votes
4answers
177 views
$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin ...
3
votes
1answer
59 views
Proof the following trig series
Prove that
$$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$
I am not necessarily looking for a ...
3
votes
2answers
56 views
Why do we need to find the intersection between these lines?
We have the functions
$$ x = -1 + 2 \cos(t)$$
$$ y = 3 + 2 \sin(t)$$
They give P's orbit
with $t$ on $\left[0, \dfrac{3}{2} \pi\right]$
Find (to 2 decimal places accurate) for which values of t ...
1
vote
2answers
34 views
Manually Finding Values of Inverse Trigonometric Functions
I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.
What is the best way to go about this? So far, I have been trying to solve the problems ...
2
votes
4answers
156 views
if $\sin24^\circ = p$ what is $\cos24^\circ$?
Let $p=\sin 24^\circ$
Then what would $\cos (24^\circ)$ be in terms of $p$?
What would $\sin (168^\circ) \cdot \sin(-78^\circ)$ be in terms of $p$?
I'm not sure how to approach these as we have ...
7
votes
2answers
137 views
$\cos(x)+\cos(x\sqrt{2})$ is not periodic
Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
0
votes
1answer
48 views
Drawing an arc between two points
I was writing a java program to draw an arc.
Arc2D.Double(int x,int y,int width,int height,int startAngle,int arcAngle,int type);
Since, I'm not familiar with the mathematics behind drawing arc, I'm ...
