Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
0
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1answer
77 views
Triangular Inequality
Let a, b, c be the three side lengths of a triangle. Prove that
$$\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\geq 3$$
Under what conditions is equality obtained?
1
vote
2answers
72 views
How do you solve $z^4 = 2(1+i\sqrt{3})$
Solve $z^4 = 2(1+i\sqrt{3})$ in the form $r(\cos\alpha+i\sin\alpha)$ where $r>0$ and $0\le\alpha<2\pi$
I know you have to find $\arctan(\frac{\sqrt{3}}{1})=\frac{\pi}{3}$ and that is $\alpha$? ...
0
votes
1answer
56 views
Euclidean triangle. Does this one exist
Does $\exists$ a Euclidean triangle $ABC$ with $\sin(A) : \sin(B) : \sin(C) = \frac{1}{4} : \frac{1}{3} : \frac{1}{2}$?
1
vote
1answer
73 views
Time Average of Cosine squared function
I've carried out the steps for the time average for $\cos^2x$ for limits $0$ to $T$.
I've gotten : $\frac{1}{T}\left[\frac{1}{2}[T+\frac{1}{4}\sin2T\right]$
I'm trying to find the average over a ...
3
votes
1answer
26 views
Largest Quadrilateral from a Set of Points
I posted the below on StackOverflow but was directed here as this may be more mathematical problem but I was looking to implement an algorithm....
I have a discrete set of points.
From this set of ...
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votes
1answer
67 views
0
votes
0answers
13 views
Is there a formula to get the changes in ship course from wind and current?
Anyone know how to get the changes of degree's in ship course that affected by wind and current?
I thinks it maybe related with the speed and degree of WIND and CURRENT. But I don't know how to ...
0
votes
2answers
26 views
Turning points on $2\sin x - x$
I'm self teaching and doing a book exercise which asks: "Considering only positive values of x, locate the first two turning points on the curve $2\sin x - x$ and determine whether they are maximum or ...
5
votes
3answers
80 views
Simplifying $\sin(2\tan^{-1} x)$
I've been working on this for a while. The answer in the book is $\frac{2x}{x^2 + 1}$ Here's my workings:
$\sin(2\tan^{-1} x)$
Let $\alpha = \tan^{-1}x \Rightarrow \tan \alpha = x$
$\sin(2\alpha) = ...
4
votes
1answer
64 views
How can I calculate the angle of a slice of an ellipse?
I'm attempting to draw a pie-chart programmatically, using an ellipse instead of a circle, but I'm having trouble calculating the correct angles for the slices. If it were a circle, I could use the ...
3
votes
1answer
64 views
Find the maximum value of $T=\frac{2}{3}(\cos 2A-\cos 2B)-\tan\frac{C}{2}$
Let $ABC$ be a triangle. Find the maximum value of
$$T=\frac{2}{3}(\cos 2A-\cos 2B)-\tan\frac{C}{2}$$
Please give me some hints. I don't know where to start
Thanks
0
votes
1answer
20 views
Coordinates of all 'N' points, equidistant from each other , on a circle of radius 'R' whose center is (h,v) from the origin?
How would I calculate the coordinates of all 'n points' equidistant from each other on a circle of radius r and the center coordinates of (h,v) from the origin .
7
votes
3answers
188 views
Broken Calculator: only certain unary functions work.
I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic ...
1
vote
2answers
45 views
Proving that $\frac {\cos(π + x)\cos(-x)}{\cos(π-x)(\frac{π}{2}+x)} = \cot^2(x)$
Prove:
$$\frac {\cos(\pi + x)\cos(-x)}{\cos(\pi - x)\cos(\frac{\pi}{2}+x)} = \cot^2(x)$$
I tried to solve the left hand side but got the answer as $-\cot(x)$ instead.
0
votes
1answer
40 views
How to find a new point on rectangle based on an known point on the same?
I have rotated a rectangle a certain amount of degree and got the point(x,y)=(130,40) which was previously (152,60). Now i want to find the x,y(marked as red) value at another location based on the ...
0
votes
0answers
35 views
Geometry question
The sides of a triangle are given to be $x^2+x+1$ , $2x+1$ and $x^2-1$. Then the largest of the three angles of the triangle is
a)75 degree
b)$\dfrac{x}{x+\pi}$
c)120 degree
d)135 degree
please ...
0
votes
1answer
44 views
Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions
Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions in terms of trigonometric functions with argument $\ln x$ multiplied by exponential function $y(x)=x^{(a+ib)x}$ or
$y(x)=x^{(a-ib)x}$
...
1
vote
1answer
29 views
Simple trigonometry simplification
$\dfrac{(1-\sec^4\theta)(1-\sin^4\theta)}{\sin^2\theta-2}$
When simplifying the expressions above, how can I eliminate the denominator?
I have tried expanding the nominators by ...
1
vote
3answers
35 views
Pythagoras/trigonometry question
Does anyone know how to answer this question?
A right-angled triangle is to be constructed with hypotenuse (the longest side) of length
one metre in such a way as to maximize the perimeter of the ...
0
votes
0answers
77 views
What is this expression called?
Could anyone please tell me if they recognize this equation? What it does is calculate the angle between two lines, but I need it's name. Any help is greatly appreciated!
$$\sin \theta = A_{1} \cdot ...
0
votes
1answer
23 views
Can I find the magnetic heading from A to B on triangle ABC if the lengths and angles inside the triangle are known?
I want to find the magnetic heading (from north) from a point $A$ to a point $B$ on a $\triangle ABC$. $A, B and C$ are moving and so the angles can be any value. as an example, the triangle could be ...
1
vote
1answer
48 views
How do I determine a formula for a given trig function?
Assume that 0 < x < pi/2 and sin(x) = z
a.) Find a formula that gives the value of sin(x/2) in terms of z
b.) Corroborate the validity of the formula for these values of x:
pi/4
pi/3
pi/6
...
4
votes
1answer
89 views
How to calculate distance on a sphere with an earth like coordination system?
This is probably a too simple question for this site, but I would really appreciate any answers.
Lets say I've got a sphere with radius $r = 70$ meters. This sphere has an Earth-like reference system ...
2
votes
2answers
76 views
A trigonometric proof
How to prove the following?
$${1-\sin A \over1+\sin A} = (\sec A- \tan A)^2$$
this is what I've done till now:
\begin{array}{ccc}
{1-\sin A \over1+\sin A} &=& {1+\sin^2 A - 2\sin A \over ...
1
vote
3answers
22 views
Mean value theorem proof with tangent
I am trying to show that:
$\tan{x}>x$ for $0<x<\pi/2$.
How can I show this? I think I can do something with the fcn $\tan{x}-x$ and it derivative, but how can I use this in a proof?
...
0
votes
1answer
39 views
Finding a function which fits this data?
I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
0
votes
3answers
27 views
steps for finding inverse tangent
My textbook gives me this:
Can someone please walk me through the steps to get -pi/3 from the inverse tangent?
1
vote
1answer
42 views
Adding integer multiples of pi
I have an angle with a given radian measurement and need to express it differently by adding integer multiples of pi. Is it accurate to say that I can simply add 4 to the coefficient of pi? It seems ...
0
votes
1answer
16 views
How can I solve an equation based off of a quadrant and equation form given an angle?
Given that 3pi/2 < z < 2pi
x = arccos(sin(z))
Given different values for z (which are angles on the unit circle) how would I write the results in these two forms, where C is a constant?:
a.) ...
2
votes
1answer
57 views
Getting an acute angle for an obtuse angle using law of Sines.
I have done this problem over and over again. I even looked up tutorials on how to properly use law of sines. It's rather embarrassing that I'm struggling so much wish this simple trigonometric stuff.
...
2
votes
2answers
67 views
Finding ALL solutions to $2(\sin^2(x)) - 5\sin(x)-3 = 0$?
What does it mean to find "ALL possible solutions?"
I know it has something to do with simplifying the equation, getting the angle (in radians) by doing the inverse.. and adding $2\pi n$?
So given ...
0
votes
1answer
35 views
Did I solve all of the steps of this Trig question properly?
Thanks to some help from the community, I think I did this problem correctly, but I would like someone to confirm that I indeed do it right. Thanks.
Question:
Let $0 \le x \le 1$.
(i.) Find the ...
1
vote
2answers
36 views
How do I write a trig function that includes inverses in terms of another variable?
It's been awhile since I've used trig and I feel stupid asking this question lol but here goes:
Given:
$z = \tan(\arcsin(x))$
Question:
How do I write something like that in terms of $x$?
Thanks! ...
3
votes
1answer
24 views
Using a particular image to justify a (specific) trig integral equality.
I would like to include the following string of equalities in a paper:
$$\sin ^2(x) + \cos ^2(x) = 1$$
$$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
1
vote
0answers
46 views
How can I find the compact trigonometric Fourier series from these signals?
I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
0
votes
1answer
51 views
Calculating the angular velocity
I have an inverted pendulum with a accelerometer mounted on the top that at rest gives me a vector up opposite to gravity, which is used to calculate the angle of the pendulum. Is it possible to ...
4
votes
5answers
133 views
Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.
Integrate $\displaystyle \int \limits_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$. I tried substituting $t=\cos x$, and then integrate with integration by parts. It got all messy... Thanks in advance for any ...
1
vote
4answers
85 views
Symmetry properties of $\sin$ and $\cos$. Why does $\cos\left(\frac{3\pi}{2} - x\right) = \cos\left(-\frac{\pi}{2} - x\right)$?
For a question such as:
If $\sin(x) = 0.34$, find the value of $\cos\left(\frac{3\pi}{2} - x\right)$.
The solution says that:
\begin{align*}
\cos\left(\frac{3\pi}{2} - x\right) &= ...
2
votes
4answers
71 views
How to find $\cos A \cos B - \sin A \sin B$?
Given that:
$\tan A=1$
and
$\tan B = \sqrt{3}$
How would you find $\cos A \cos B - \sin A \sin B$?
EDIT: This is what I've tried after reading bhattacharjee's answer:
$$ \tan(A+B) = \tan A+\tan ...
3
votes
1answer
100 views
How to simplify this trigonometric expression?
I was trying to solve a problem taken from an Physics Olympiad when I came across a curious and complex mathematical expression. I can not prove with what I know so far about mathematics, does could ...
0
votes
1answer
45 views
Find all solutions/zeros
Find all solutions in the interval$ [0,2\pi]$:
1)$3\sec^3x=4\sec x$
2)$2\sin^2x+3\sin x+1=0$
3)$\cos x+\sin x \tan x=0$
4)$2\sec^3x+\sec^2x–8\sec x–4=0$
Thnx so much
1
vote
1answer
59 views
Calculating mean velocity of an orbiting body as it moves towards a point.
I'm making a game, in the game planets orbit a central point in circular orbits, they move directly towards their targets and the vector is simply added to their orbital path. Whilst not realistic it ...
1
vote
1answer
67 views
Figuring the volume of a partially filled cone without the radius of the material inside the cone
Say I have a cone. For simplicities sake, this cone is at the bottom of a storage silo. Is has a flat bottom, flat top and angled sides. I know the height of the material inside the cone but don't ...
1
vote
1answer
16 views
Direction and Distance to a line which goes from location A to location B
I have two location specified in latitude and longitude. Between these location is a line (bearing). I am at a third location. How do I determine how far I am away from this line and in what direction ...
5
votes
1answer
88 views
Expressing a number in $\sqrt a/b$ form
Express the number $\sqrt3 \sin(10^\circ) +\dfrac38\tan(10^\circ)$ in the form $\dfrac{\sqrt a} b$, where $a$ and $b$ are integers.
I am sure that trigonometric formulas must be used here, but I ...
2
votes
3answers
61 views
Incoherence using Euler's formula
Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
2
votes
0answers
35 views
Bernoulli generating function and cotangent
May I ask for a little help in solving a problem about Bernoulli number generating function?
Bernoulli number generating function is given by: $$f(z):=\begin{cases}
\frac{z}{e^{z}-1} & z \in ...
0
votes
3answers
38 views
Trigonometric proof involving sec and cosec
How should I proceed to prove the following:
$$\sec^2 A +\csc^2 A = \sec^2 A \cdot \csc^2 A$$
This is what I've attempted:
$$LHS = 1 + \tan^2 A + 1 + \cot^2= 2+ 1\cot^2 + \cot^2$$
However, this ...
1
vote
3answers
87 views
How to find $\cos A$ from $\sec$ and $\tan$?
It's given that
$$\sec A + \tan A = 4$$
How would you find $\cos A$ from this?
1
vote
1answer
19 views
Formula for Damage Calculation Based on Distance
I am working on programming a game, and I want the damage that a player takes to be based off of the distance they are from the explosive. Consider it like a grenade. Right on the blast radius ...
