Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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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
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1answer
29 views

trigonometric inequality - how to prove it?

Let $ 0 < x < \frac {\pi}{2}$ How to prove it? $$2 \sin x \le x- \frac {\pi}{3} + \sqrt {3} $$
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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?
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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
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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 ...
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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
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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
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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 ...
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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.
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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
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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
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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: $A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
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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
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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 $n \in \mathbb{N}$?
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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 ...
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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
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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 ...
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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 ...
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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 & ...
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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
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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
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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
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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
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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?
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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
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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
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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 ...
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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?
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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
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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: $$\int \frac{(x^2+3x+4)\,dx} {\sqrt{x^2-4x}}$$
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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
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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 ...
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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) = ...
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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
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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 ...
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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 ...
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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
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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
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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 ...
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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 ...
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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
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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
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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
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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 ...

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