Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
0
votes
1answer
64 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
50 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
33 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
49 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:
$$
...
1
vote
0answers
29 views
Can you help me reverse the Minimum Curvature Method?
The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
2answers
32 views
Integral of $\int \frac{\sin(x)dx}{3-\cos(x)}$
I am trying to solve this integral and I need your suggestions.
I don't know if its OK to set $3-\cos(x)$ as $t$ $\rightarrow dt = \sin(x)dx$ or just take $\cos(x)$ and set it as $t$
$$\int ...
2
votes
4answers
64 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 ...
13
votes
3answers
162 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
16 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
362 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
43 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
66 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:
$$f'(x) = \dfrac{\sin(x) \cdot \dfrac{2}{\cos^2(2x)} - \tan(2x) \cdot ...
6
votes
2answers
44 views
relationship of polar unit vectors to rectangular
I'm looking at p. 16 of Fleisch's Student's Guide to Vectors & Tensors. He's talking about the relationship between the unit vector in 2D rectangular vs. polar coordinate systems. He gives these ...
2
votes
3answers
42 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
0
votes
0answers
63 views
Solution procedure for a system of trigonometric equations in two variables
i would like to know if there's a method for solving the following system using (or not) tan half angle substitution.
$$A\cdot\sin(\theta_1) + B\cdot\cos(\theta_1) + C\cdot\sin(\theta_3) + ...
3
votes
3answers
102 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
85 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
65 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.
9
votes
1answer
95 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?
16
votes
4answers
207 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:
$$\int\frac{\cos x}{\sin ...
3
votes
1answer
63 views
Proof the following trig series
Prove that
$$\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}$$
I am not necessarily looking for a ...
3
votes
2answers
57 views
Why do we need to find the intersection between these lines?
We have the functions
$$ x = -1 + 2 \cos(t)$$
$$ y = 3 + 2 \sin(t)$$
They give P's orbit
with $t$ on $\left[0, \dfrac{3}{2} \pi\right]$
Find (to 2 decimal places accurate) for which values of t ...
1
vote
2answers
36 views
Manually Finding Values of Inverse Trigonometric Functions
I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.
What is the best way to go about this? So far, I have been trying to solve the problems ...
2
votes
4answers
161 views
if $\sin24^\circ = p$ what is $\cos24^\circ$?
Let $p=\sin 24^\circ$
Then what would $\cos (24^\circ)$ be in terms of $p$?
What would $\sin (168^\circ) \cdot \sin(-78^\circ)$ be in terms of $p$?
I'm not sure how to approach these as we have ...
7
votes
2answers
144 views
$\cos(x)+\cos(x\sqrt{2})$ is not periodic
Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
0
votes
1answer
53 views
Drawing an arc between two points
I was writing a java program to draw an arc.
Arc2D.Double(int x,int y,int width,int height,int startAngle,int arcAngle,int type);
Since, I'm not familiar with the mathematics behind drawing arc, I'm ...
8
votes
2answers
166 views
A hard 'if and only if' trigonometric identity proof
Prove
$$
\frac{-2+2\tan A+2\cos B\cdot\sin B+\cot^2 A\cdot({\sec^4A-\operatorname{cosec}^2A-2)}}{2+\tan^2A-2\sin^2A} =(\sin A+\cos A)^2
$$
if and only if B is the double angle of A, or ...
0
votes
1answer
42 views
Why is it necessary to use sin or cos to determine heading? (dead reckoning)
Here's the problem:
(see pic for problem): https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-ash3/21281_10152793202590262_1804321932_n.jpg
You have a robot that is moving forward at a variable rate ...
7
votes
1answer
95 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
2
votes
2answers
67 views
How can I prove this cosine equation?
How to prove that $\cos(90)\cos(\theta)+\sin(90)\sin(\theta)=\sin(\theta)$ ?
3
votes
1answer
44 views
Show that the following product equals 1 (involves trig)
How can I show that:
$$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$
Could you please explain to me how to approach this problem?
Thank you.
2
votes
3answers
66 views
Are the names and symbols for common mathematical operators the same in every language?
Do all human languages that have arithmetic use +, -, ×, ÷, and ^?
How about sin, cos, tan, asin, acos, and atan?
1
vote
1answer
42 views
prove this trigonometric expression
If $$\tan \theta +\sin \theta =m $$
and
$$\tan \theta -\sin \theta =n$$
then prove that $$m^2-n^2=4\sqrt{mn}$$
I've tried to $(m^2-n^2)$ as $(m-n)(m+n)$ but can't get to RHS.
3
votes
2answers
44 views
what is the value of this trigonometric expression
I want to find out value of this expression
$$\cos^2 48°-\sin^2 12°$$
Just hint the starting step.Is there any any formula regarding $\cos^2 A-\sin^2 B$?
2
votes
1answer
14 views
value of this inverse trigonometric expression.
How to evaluate this expression.
$$\sec^2(\tan^{-1} 2)+\csc^2(cot^{-1}(3))$$
I'm stuck on how to process squares, which is on sec and cosec function?.
10
votes
2answers
258 views
A matrix w/integer eigenvalues and trigonometric identity
Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated:
Let $n$ be a natural number.
(a) Consider the following Toeplitz/circulant symmetric matrix:
...
1
vote
2answers
59 views
Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?
The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions).
So, ...
0
votes
1answer
40 views
Trigonometry - Addition and subtraction theorem
If $\theta$ and $\phi$ are angles between $0°$ and $90°$, and $\sin \theta=3/5$ and $\tan \phi=7/24$, find without the use of a calculator, the value of each of the following:
a. $\sin(\theta−\phi)$
...
1
vote
2answers
37 views
Derivative of Trig Functions (Intuition Help?)
Looking for some intuition help here.
I have the following exercise and these are the steps I take:
$$
y = \sin\left(\frac{1}{x}\right)
$$
$$
u=\frac{1}{x}
$$
$$
y = \sin u,\;\;\frac{dy}{du} = \cos ...
17
votes
1answer
181 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
0
votes
0answers
33 views
Get the entrance point from a straight line in a rectangle
The rectangle is like a street. The right half is to go upwards, the left half to go down. The red lines are paths of vehicles. And my goal is to give every vehicle the right lane. So when you look at ...
0
votes
1answer
67 views
How to find a point on the tangent line whos length is 1?
im trying to figure out a formula to find the point(x,y) on a tangent line whos length is between 0 and 1 while it rotates around the unit circle uniformly, so the point would either be right on the ...
0
votes
0answers
49 views
Trigonometric bearing problem
I have two trigonometric problems that I solved, however it does not match the answer in the book:
1) A yacht crosses the start line of a race on a bearing of $31$ degrees. After $4.3$ km, it ...
1
vote
2answers
60 views
How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $
How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
12
votes
2answers
122 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
