Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
3
votes
3answers
79 views
Finding trigonometric integral (challenging)
Integrate:
$$\displaystyle\int \dfrac{(1+\cos^2A\times\cos(2A))^2}{\cos(2A)\times(2\cos^4A+\sin^2A)}dA$$
Again my working with trigonometric identities has not take me anywhere useful, that is ...
3
votes
2answers
57 views
Trigonometry Airplane question. Finding bearing and distance.
A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
0
votes
0answers
33 views
Specular intensity
Im currently studying for an exam and have been going through some past papers on the subject, however i have come across a question that has recursively come up each year and the notes on it are not ...
1
vote
2answers
84 views
How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?
How do i prove that $\frac{1}{\pi} \arccos(1/3)$
is irrational?
1
vote
0answers
36 views
Basic Trigonometric Substitution Question
I have a basic trig substitution question for integrals. It always seems that x is opposite to the theta angle. However, making x the adjacent on the right angle triangle seems to work just fine as ...
4
votes
5answers
76 views
$\lim_{x\to \pi/2} \;\frac 1{\sec x+ \tan x}$
how to solve it answer is $0$, but $\frac 1{\infty + \infty}$ is indeterminate form
$$\lim_{x \to \pi/2} \frac 1{\sec x + \tan x}$$
1
vote
0answers
34 views
simplification of a natural log of a trigonometric function
hope you are all well.
I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression:
$$k\cos X \cdot 4\ln(\cos X)$$
where $k$ is a constant and $X$ ...
1
vote
2answers
42 views
Identity for a weighted sum of sines / sines with different amplitudes
I'm trying to simplify the following sum of sines with different amplitudes
$$
a \sin(\theta) + b \sin(\phi) = ??? \,\,\,\,\, (1)
$$
I know that
$$
a \sin(\theta) + a \sin(\phi) = ...
2
votes
2answers
70 views
Prove $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $
Prove that $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $ where $\alpha$ $,\beta$ $,\gamma$ are the angles of a triangle
0
votes
0answers
21 views
Discretizing a cosine function?
I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources:
$$f(k,x) = \sqrt2 ...
2
votes
4answers
43 views
How can I solve this Laws of Sines problem?
This is a homework question that was set by my teacher, but it's to see the topic our class should go over in revision, etc.
I have calculated $AB$ to be 5.26m for part (a). I simply used the law ...
2
votes
1answer
26 views
How to Find the Center of a Parallelogram
I want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It seems ...
4
votes
2answers
155 views
If ${ x }^{ 4 }+{ y }^{ 2 }=1$ then $x$ and $y$ can be both rational numbers?
Can you give two numbers $(x,y)\in\mathbb{Q}$ such that ${ x }^{ 4 }+{ y }^{ 2 }=1$.
I don't know if exists or not. I derive this equation questioning that if $\sin { \alpha } ={ x }^{ 2 }$ for ...
0
votes
3answers
64 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
4
votes
1answer
33 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
56 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 $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
5
votes
3answers
101 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
40 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
41 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
47 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
31 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) = ...
6
votes
2answers
95 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
32 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
40 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
46 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
56 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
74 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
72 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
86 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
106 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
50 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
30 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
38 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
63 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
37 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
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:
$$
...
1
vote
0answers
23 views
Can you help me reverse the Minimum Curvature Method?
The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
2answers
31 views
Integral of $\int \frac{\sin(x)dx}{3-\cos(x)}$
I am trying to solve this integral and I need your suggestions.
I don't know if its OK to set $3-\cos(x)$ as $t$ $\rightarrow dt = \sin(x)dx$ or just take $\cos(x)$ and set it as $t$
$$\int ...
2
votes
4answers
62 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 ...
12
votes
3answers
149 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
355 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
42 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
65 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
43 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
40 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{
& ...
