Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
1
vote
2answers
20 views
Determine depth of a partially filled hemisphere
Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer.
The question goes like this:
A bowl ...
1
vote
1answer
29 views
trigonometric inequality - how to prove it?
Let $ 0 < x < \frac {\pi}{2}$
How to prove it?
$$2 \sin x \le x- \frac {\pi}{3} + \sqrt {3} $$
0
votes
2answers
35 views
Exponential trigonometrical equation
Find $x$ from $[-\frac{\pi}{2},\frac{\pi}{2}]$ in: $$2^{\sin 3x}-8^{\sin x}=\sin^3{x}$$
I know that $x=0$ verifies the equation, but is it the only solution?
0
votes
0answers
14 views
Calculating inner angles, length, area of polygon from gps coordinates
I've a set of polygons. Each polygon is described by 4 Points (Longitude,Latitude).
How can I calculate the area of the polygon, the inner angles of each angle and the length of all sides and finally, ...
7
votes
2answers
103 views
How do I solve this inequality? $\sin x < 2x^3$
The equation is
$\sin x < 2x^3$
The steps I've taken so far are:
$\sin x < 2x^3 $
$\sin x - 2x^3 < 0 $
To solve this I should find when the slope is $ 0 $ so I can find the max and ...
0
votes
1answer
29 views
Why is the length R cosine theta?
Why is the length described as R cosine theta (the top where the Sphere is sliced off)? I've been staring at the geometry for quite a bit & can't figure.
Thanks
3
votes
5answers
111 views
Graphing $\sin(|x|)$?
I'm confused on how the graph is in quadrant II and III. If $|x|$ is evaluated first wouldn't all the answers be positive, so that when the range of $|x|$ is plugged into $\sin$ wouldn't the range of ...
5
votes
3answers
105 views
Solve $\tanα+2\tan2α+4\tan4α+8\tan8α+16\tanα=\cotα$ for $\alpha$
My knowledge of trigonometry are still insufficient to resolve this problem. Any help would be greatly appreciated.
Solving for $\alpha$:
$$\tanα+2\tan2α+4\tan4α+8\tan8α+16\tanα=\cotα$$
6
votes
1answer
60 views
How prove this $\frac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\frac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\frac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$
let $0<A,B,C<\dfrac{\pi}{2}$,and $A+B+C=\pi$,prove that
$$\dfrac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\dfrac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\dfrac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$$
my ...
1
vote
2answers
40 views
How to calc arc sine without a calculator?
How can I find the arc sine of a sine without using a calculator? Thank you.
-3
votes
1answer
45 views
$k = $k=tan (27θ) - tanθ $ and $h = \frac {sin(θ)}{cos(3θ)} + \frac {sin(3θ)}{cos(9θ)} + \frac {sin(9θ)}{cos(27θ)} $ [closed]
If $k = tan (27θ) - tanθ $ and
$h = \frac {sin(θ)}{cos(3θ)} + \frac {sin(3θ)}{cos(9θ)} + \frac {sin(9θ)}{cos(27θ)}
$
then prove $k =2h$
6
votes
3answers
100 views
How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
$$\int \dfrac{1}{\cos x+\tan x }dx$$
This can be converted to
$$\int \dfrac{\cos x}{\sin x+\cos^2x}dx$$
But from here I get stuck. Using t substitution will get you into a mess. Are there ...
0
votes
1answer
88 views
Evaluation of the integral $\int \cos\omega t\ln\cos\omega t\,dt$
I am trying to evaluate an integral of the form
$$ \int \cos\left(\omega t\right) \ln \cos\left(\omega t\right) dt$$
and am unsure how to proceed.
I rewrote it as:
$$ \textrm{Re} \left\{\int dt ...
0
votes
1answer
65 views
Evaluate the maximum of: $A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
Given a triangle ABC. Evaluate the maximum of:
$A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
0
votes
2answers
48 views
Orthogonality of eigenvectors of laplacian
Let $x_i=(\sin i\pi/n,\cdots,\sin (n-1)i\pi/n)$ for $i=1,\cdots,n-1$. I want to show that $x_i \cdot x_j=\delta_{ij} n/2$. Why is it true? I tried $\sin a \sin b=-[\cos(a+b)-\cos(a-b)]/2$ but don't ...
2
votes
1answer
25 views
Inverse Trigonometry doubt.
Suppose $\sin y=\sin 2x$, then what will be the solution for $y$?
Will it be $y=2x$ or $y=n\pi-2x$ for some $n \in \mathbb{N}$?
3
votes
4answers
48 views
Find maximum value of $f(x)=2\cos 2x + 4 \sin x$ where $0 < x <\pi$
Find the maximum value of $f(x)$ where
\begin{equation}
f(x)=2\cos 2x + 4 \sin x \ \
\text{for} \ \ 0<x<\pi
\end{equation}
3
votes
3answers
100 views
$\int \frac{1}{\cos(x)}\,\mathrm dx$
could you help me on this integral ?
$$\int \frac{1}{\cos(x)}\,\mathrm dx$$
Here's what I've started :
$$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int ...
0
votes
1answer
64 views
evaluate the following limit on trigonometry
given that \begin{equation}
\lim_{y \rightarrow 0}
\frac{\sin y}{y}=1
\end{equation}
evaluate the following
\begin{equation}
\lim_{x \rightarrow 0}
\frac{2-2\cos^2 x-2 \cos x \sin ^2 x}{x^4}
...
2
votes
1answer
60 views
minimum value of a trigonometric equation is given. the problem is when the minimum value attains
Suppose the minimum value of $\cos^{2}(\theta_{1}-\theta_{2})+\cos^{2}(\theta_{2}-\theta_{3})+\cos^{2}(\theta_{3}-\theta_{1})$ is $\frac{3}{4}$.
Also the following equations are given
...
1
vote
1answer
46 views
Rotation angle of regular polygon that has largest taxicab maginitude between all vertices
Firstly just to apologise, I posted this on mathoverflow before realising it was focused on research level mathematics.
If I have a regular polygon that is centred at the origin. Then take the ...
1
vote
3answers
47 views
Given $y=\arccos(x)$ find $\arcsin(x)$ in terms of y
Given that $y = \arccos x$, $ - 1 \le x \le 1\,and\,0 \le y \le \pi $, express $\arcsin x$ in terms of y.
The best I know how to do this is is:
$$\eqalign{
& \cos y = x \cr
& ...
1
vote
2answers
25 views
Verifying the trigonometric identity $\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$
I have the following trigonometric identity
$$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$
I've been trying to verify it for almost 20 minutes but coming up ...
6
votes
4answers
117 views
Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
Testing with my calculator in degree mode, I have found the following to be true:
$$\tan \left(90 - \frac{1}{10^n}\right) \approx \frac{180}{\pi} \times 10^n, n \in \mathbb{N}$$
Why is this? What is ...
0
votes
1answer
21 views
JQuery placing elements X pixels by degree (Basic trig)
Essentially I want to place an element X pixels from the current position towards the center. Here's my code:
...
2
votes
1answer
63 views
The angle $\theta$ lies in Quadrant IV with point $P$ on the terminal arm and $\tan\theta=-\frac{3}{5}$?
The angle $\theta$ lies in Quadrant IV with point $P$ on the terminal arm and $\tan\theta=-\dfrac{3}{5}$?
My friend explained that.
I'm not sure if he is correct.
In Quadrant IV the $\sin\theta$ and ...
3
votes
5answers
66 views
Is it possible to find the sine or cos from a tangent?
I have a value of a tangent. Is it possible to find the sine and/or cossine from that value? How?
0
votes
3answers
18 views
Coordinates of Intersection of two circles
i am trying to find the coordinates of the intersection points of two circle.
Given value is the center coordinates and radius of both the circle
Please help without using equation substitution ...
2
votes
3answers
55 views
What 's the differece between $\cot(x)$ and $\arctan(x)$? [duplicate]
I know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$
What is the difference between these two function?
Is $\cot(x)$ the ...
4
votes
1answer
51 views
Calculating the Roots of Sine
Aside from the obvious knowledge that the roots of $\sin x$ are all integer multiples of $\pi$, is there a formal, algebraic method to calculate the roots of trigonometric functions similar to the ...
1
vote
2answers
35 views
Right Triangle Trig
I need to find the measure of each angle indicated and round to the nearest tenth.
I am given two sides 12 and 13 and one angle C which is 90 degrees. How do I figure this out?
0
votes
1answer
36 views
Calculating Trigonometric Ratios for Sine and Cosine
The Sine, Cosine of x can be computed as follows:
$$\sin(x) = x - \dfrac {x^3}{3!} + \dfrac {x^5}{5!} - \dfrac {x^7}{7!} + \dfrac {x^9}{9!} …$$
$$\cos(x) = 1 - \dfrac {x^2}{2!} + \dfrac {x^4}{4!} - ...
3
votes
5answers
66 views
Help with a trig-substitution integral
I'm in the chapter of trigonometric substitution for integrating different functions. I'm having a bit of trouble even starting this homework question:
$$\int \frac{(x^2+3x+4)\,dx} {\sqrt{x^2-4x}}$$
7
votes
3answers
86 views
Given that $x = 4\sin \left( {2y + 6} \right)$ find dy/dx in terms of x
My attempt:
$\eqalign{
& x = 4\sin \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = \left( 2 \right)\left( 4 \right)\cos \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = 8\cos ...
1
vote
6answers
59 views
Product-to-sum formulas?
My old pre-calculus book says:
$$\sin u\cos v=\frac{1}{2}[\sin (u+v)+\sin(u-v)]$$
and $$\cos u \sin v=\frac{1}{2}[\sin(u+v)-\sin(u-v)]$$ I don't understand why there is a difference, since ...
1
vote
1answer
40 views
Trig identity $\sin(x)\cos(x) = \sin(2x)/2$?
http://tutorial.math.lamar.edu/Classes/CalcIII/TICylindricalCoords_files/eq0014MP.gif
Could someone tell me how $\sin(2x)$ arrives?
I know that there is a trig identity that says $2\cos(x)\sin(x) = ...
1
vote
1answer
32 views
Trigonometry and computations - what to do?
I want to prove the following equality:
$$\frac{\sin3x}{\sin x}\cdot\frac{\sin(2n+1)x}{\sin x}=\frac{\sin(2n-1)x}{\sin x}+\frac{\sin(2n+1)x}{\sin x}+\frac{\sin(2n+3)x}{\sin x}$$
I don't know which ...
3
votes
1answer
33 views
Is $S=\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$ finite?
Problem:
If $$S=\sum_{r=1}^\infty \tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ Then find S ??
Solution:
I know that $\tan^{-1} x + \tan^{-1} y= \tan^{-1} \frac {x +y} {1-xy} $
But I have no idea ...
2
votes
1answer
26 views
The angle at which a circle and a hyperbola intersect?
$x^2 - 2y^2 = 4$
$ (x-3)^2 + y^2 = 25 $
How do you calculate the angle at which a circle and a hyperbola intersect?
If I express $y^2$ from the first equation and apply it to the second ...
1
vote
2answers
51 views
Why do we use the inverse conversion formula to convert slope per radians to slope per degrees
This is a contribution question I'm making in hopes that others may benefit. I will provide my answer underneath. Initially I wanted to ask this question, but I solved it myself and I'd like to give ...
1
vote
1answer
41 views
For the slope of the line at a point, why am I getting a different result by using the calculus method?
I am evaluating the slope of the secant as it approaches $f(30)$ for the function
$f(x) = 2\sin(x) - 2$.
Using calculus I can easily find that the derivative is $f'(x) = 2\cos(x)$.
If I sub in $30$ ...
1
vote
0answers
25 views
Non-linear time-dependent equation
Given
$$u(x,t)=\cos(t)\cos\left(5 \pi (x-1)/2\right)$$
is the actual solution to
$$u_{t}=\epsilon^{2}u_{xx}+(1-u^{2})u+f(x,t)$$
I want to find what f(x,t) should be. I've simplified the results to ...
1
vote
0answers
28 views
Second order linear ODE with trigonometric coefficient
Is there a theory and a name for the second order linear ODE with trigonometric coefficient (other than the Floquet theory)?
The equation in question, with $a$,$b$,$c$ periodic function containing ...
0
votes
1answer
39 views
How to find the sine of an angle
How to find the sine/cos/tangent/cotangent/cossec/sec of an angle:
In degrees
$\sin(23^{\circ}) =$ ?
In radians
$\sin(0.53) =$ ?
4
votes
2answers
64 views
Ratio between trigonometric sums: $\sum_{n=1}^{44} \cos n^\circ/\sum_{n=1}^{44} \sin n^\circ$
What is the value of this trigonometric sum ratio: $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ} = \quad ?$$
The answer is given as ...
4
votes
1answer
64 views
Maximum and Minimum Value of $f(x)$
$$f(x)=\sin(x)+\int_{-\pi/2}^{\pi/2}\left(\sin(x)+t\cos(x)\right)f(t)\,\mathrm dt$$
Find maximum and minimum values of $f(x)$.
I tried to simplify this expression by checking even or odd ...
2
votes
1answer
91 views
Question about $\int_\Omega\!\cos^n\alpha\cdot\cos\theta_o\,d\omega_o$
I see this integral metioned in this paper (at the start of section 3.2 ,p.4)
$$\int_\Omega\!\cos^n\alpha\cdot\cos\theta_o\,d\omega_o$$
It's an integral over hemisphere and the $\alpha$ term means ...
0
votes
3answers
46 views
Trig word problem.
There is a circular pen with a goat in it.
The goat is tethered by a rope to the edge of the pen.
The rope is the length of the radius of the pen.
What area of grass in the pen can the goat graze?
1
vote
1answer
43 views
Finding the x-coordinate of the max point of $y = x\sqrt {\sin x} $ so that it satisfies the equation $2\tan x + x = 0$
The maximum point on the curve with equation $y = x\sqrt {\sin x} $, $0 < x < \pi $, is the point A, Show that the x-coordinate of point A satisfies the equation $2\tan x + x = 0$
I ...
0
votes
1answer
34 views
What was the initial velocity in the y direction vx = 3.6 m / s * cos 18 °?
A ball is thrown, it's path is oblique; it's velocity is depicted by the $x$-axis formula $v_x = 3.6 \text{ m/s} \times \cos 18^{\circ}$. What was the initial velocity in the $y$ direction?
I have no ...
