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

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1k views

Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]

For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
4
votes
4answers
860 views

Evaluate $\tan ^{2}20^{\circ}+\tan ^{2}40^{\circ}+\tan ^{2}80^{\circ}$

Evaluate $\tan ^{2}20^{\circ}+\tan ^{2}40^{\circ}+\tan ^{2}80^{\circ}$. Can anyone help me with this? Thank You!
0
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1answer
136 views

Need a formula for distance based on scale

How can I calculate the distance to an object of known size in a photo using scale (I do not have any camera lens info, and will likely need to perform this calculation several times on different ...
4
votes
3answers
217 views

Prove $\frac{1+\cos{(2A)}}{\sin{(2A)}}=\cot{A}$

I am sorry to ask so many of these questions in such as short time span. But how would I prove this following trigonometric identity. $$ \frac{1+\cos(2A)}{\sin(2A)}=\cot A $$ My work thus far is $$ ...
0
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4answers
548 views

Prove $\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$

How would I prove the following double angle identity? $$\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$$ My work thus far is $$\frac{1-2\sin A\cos A}{\cos^2A-\sin^2A}$$ $$\frac{1-2\sin ...
0
votes
4answers
560 views

Prove $\cot A\sin 2A=1+\cos 2A$

How would I prove the following two trigonometric identity. $$\cot A\sin 2A=1+\cos 2A$$ This is my work so far $$\frac{\cos A}{\sin A}(2\sin A \cos A)=1+\cos 2A$$ I am not sure what I would do ...
0
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2answers
331 views

Prove $\frac{\sin(A+B)}{\cos(A-B)}=\frac{\tan A+\tan B}{1+\tan A\tan B}$

How would I solve the following double angle identity. $$ \frac{\sin(A+B)}{\cos(A-B)}=\frac{\tan A+\tan B}{1+\tan A\tan B} $$ So far my work has been. $$ \frac{\sin A\cos B+\cos A\sin B}{\cos A\cos ...
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6answers
3k views

Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B) $$But I am not sure ...
10
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2answers
317 views

Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals?

(My personal "feel" is that $\sin$ and $\cos$ are first-class citizens, $\tan$ is "1.5th-class," and the rest are second-class; I'm sure there are others who feel the same.) Main question(s): From a ...
5
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1answer
137 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
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2answers
127 views

A sum of terms like $(1 + \sin^2x)^{-1}$

How to prove that \[ \frac12\frac1{1+\sin^2 x} + \frac12\frac1{1+\cos^2 x} + \frac12\frac1{1+\sec^2 x}+ \frac12\frac1{1+\csc^2 x} = 1? \] Some genius please help me I have been stuck at this for one ...
1
vote
2answers
456 views

Formula to calculate the height of a Satellite Image in degrees

Do you know of a formula or function that can calculate the real world height of a satellite image in degrees? The image is from Google Static Maps and the image uses Mercator projection which makes ...
15
votes
4answers
431 views

Inequality for cosines

Is the following inequality in a triangle known? $$4(\cos A + \cos B + \cos C) \le 3 + \cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$$ It looks ...
2
votes
1answer
205 views

Trigonometric inequality for angles in triangle

Let $A, B, C$ be angles in a triangle. Is the following inequality $$4\cos A \le 1 + \cos\left(\frac{B-C}{2}\right)$$ true? I just assume it but don't have a proof. Thank you for your help.
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3answers
2k views

Finding the values of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.

i know that the values of $\cos n\pi=(-1)^{n}$ and $\sin n\pi=0$. Now i want to know that what is the general expressions of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.
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1answer
202 views

Definite integral with a complex number in Euler form

Well... I spent an hour trying to figure out how to go from lhs to rhs: $$\frac { 1 }{ 2\pi } \int _{ -\infty }^{ +\infty } \phi _{ T }(u)\left( \int _{ k }^{ +\infty } e^{ -iux }dx \right) ...
1
vote
1answer
42 views

Finding the values of $Q$ from $180^{\circ}$ to $360^{\circ}$

I have $\cos{Q}=-0.5$, required to find $180^{\circ}\leq Q\leq 360^{\circ}$. What I tried: The acute angle is $60^{\circ}$, then, since cosine is negative on third quadrant, I suppose the value is ...
21
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2answers
569 views

an integer sum of products of tangents

This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that ...
36
votes
4answers
773 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
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4answers
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Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...
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3answers
468 views

Graph $y=1/\sin x$

Graph $y=\dfrac{1}{\sin x}$ Now, I looked at the graph on google and got this Which I thought that $y=\dfrac{1}{\sin x}$ would be $y=\sin^{-1}x$ But it's apparently not. So if anyone can shed ...
4
votes
0answers
371 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
2
votes
2answers
101 views

Prove $ (r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$

How would I verify the following trig identity? $$(r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$$ My work thus far is $$(r^2\cos^2A\sin^2A)+(r^2\sin^2A\sin^2A)+(r^2\cos^2A)$$ But how would ...
1
vote
1answer
39 views

Finding more details about a triangle using the given details.

In the triangle $ABC$ we have $\tan{\frac{A}{2}}=\frac{1}{3}$ $b+c=3a$ Specify which of the following answers is correct: $a) m(\angle B)=\frac{\pi}{2}$ or $m(\angle C)=\frac{\pi}{2}$ $b) ...
4
votes
3answers
248 views

Solving $E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$

$$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$ I got no idea how to find the solution to this. Can someone put me on the right track? Thank you very much.
3
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2answers
122 views

Trigonometric equation inversion

I am trying to invert the following equation to have it with $\theta$ as the subject: $y = \cos \theta \sin \theta - \cos \theta -\sin \theta$ I tried both standard trig as well as trying to ...
1
vote
2answers
184 views

What is the answer for the $\lim\limits_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}$?

Let $t\in (0,\pi)$ and $n$ change in natural numbers. I am wondering what is the answer to the following limit. $$\lim_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}.$$ Thank you.
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2answers
43 views

Find the possible value of $∠Z$ in the following case.

There's a triangle $XYZ$, where $X, Y, Z$ are the angles and $x, y, z$ are the sides opposite to the angles respectively. Given that $2X = 3Y$ and $x = 2z$, then find the possible value of ...
4
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0answers
254 views

Reprojecting/converting an orthographic image/grid into a cartesian grid

I'm trying to dewarp a fisheye image into a simple rectilinear image of a subset of the fisheye. As part of this, I'm trying to map the azimuth/altitude values into a point on the image. The points ...
5
votes
3answers
647 views

Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$

how to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$? I know how to prove $ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $ in this way: $ \tan{20^0} ...
2
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2answers
116 views

Verify $(2r\sin A\cos A)^2+r^2(\cos^2 A-\sin^2 A)^2=r^2$

How would I verify this confounding identity: $$(2r\sin A\cos A)^2+r^2(\cos^2 A-\sin^2 A)^2=r^2.$$ I know that $$\sin 2\theta = 2\sin \theta \cos \theta$$ and that $$\cos^2 \theta - \sin^2 \theta ...
1
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2answers
490 views

Verify $\frac{\sin^3A + \cos^3A}{\sin A + \cos A} = 1 - \sin A\cos A$

How can I verify the following trigonometric identity? $$\frac{\sin^3 A + \cos^3 A}{\sin A+\cos A} = 1-\sin A\cos A.$$ My work so far is $$\begin{align*} ...
13
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4answers
713 views

$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral

Here is a fun integral I am trying to evaluate: $$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$ I thought about integrating by parts $2n$ times and then using ...
3
votes
2answers
124 views

Formula obtained by using Trignometric approximation for a triangle with a very small side

I am reading a paper on the force between hooft polyakov monopoles, but I am completely baffled by one of the 'elementary trignometric' equation they have got using an approximation. Consider a ...
0
votes
2answers
2k views

How to calculate the displacement between points?

I'm having trouble finding the right formula for displacement between two points. I'm working on a program that will place a digital ruler and allow the user to trace their finger on the edge of the ...
3
votes
5answers
271 views

Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$

Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$ $$\tan x+\sec x=2\cos x$$ $$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$ $$\left(\dfrac{\sin ...
4
votes
2answers
164 views

Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$

Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$ $$\cos^2x-\sin^2x=\sin$$ Edit $$1-\sin^2x-\sin^2x=\sin x$$ $$2\sin^2 x+\sin x-1=0$$ $\sin x=a$ $$2a^2+a-1=0$$ $$(a+1)(2a-1)=0$$ ...
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3answers
92 views

Prove $\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}$

How would I simplify this difficult trigonometric identity: $$\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}.$$ I am not exactly sure what to do. I simplified the right side ...
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votes
3answers
246 views

$3\sin^2x=\cos^2x;$ $ 0\leq x\leq 2\pi$ Solve for $x$

$3\sin^2x=\cos^2x;$ $0\leq x\leq 2\pi$ Solve for $x$: I honestly have no idea how to start this. Considering I'm going to get a number, I am clueless. I have learned about $\sin$ and $\cos$ but I ...
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3answers
184 views

Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$

Can anyone help me solve the following trig equations. $$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$ My work thus far ...
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votes
1answer
241 views

Solve $\ddot\theta +k\sin(2\theta)=0$ given initial value and constraints

How is it possible to deduce from the equation $$\ddot\theta +k\sin(2\theta)=0$$ where $\theta=\theta(t)$ and $\tan(\theta)={b(t)\over a(t)}$, $k$ is constant, and $a(0)=a_0$, $a(t)^2+ b(t)^2=a_0^2$. ...
2
votes
1answer
73 views

$k$ in trigonometric equality $\sin(a) =\sin(b)$

On a test there is the question: "Solve for $x$ on the interval $[-\pi,\pi]$ where $\sin(2x) = \cos(3x)$ I know that: $\cos(x) = \sin(\frac12\pi - x)$ So you can rewrite the equation to: ...
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votes
3answers
2k views

Is there any way to find a angle of a complex number without a calculator?

Transforming the complex number $z=-\sqrt{3}+3i$ into polar form will bring me to the problem to solve this two equations to find the angle $\phi$: $\cos{\phi}=\frac{\Re z}{|z|}$ and ...
0
votes
5answers
166 views

Verify trigonometry equation $\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A$

Sorry for asking so many of these type of questions. How would I verify the following trigonometry identity: $$\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A.$$ My work is $$\frac{\sin A + ...
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2answers
72 views

Verify trigonometry equation $\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$

How would I verify the following trig equation? $$\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$$ My work so far is to write the RHS as $$\frac{1/\cos(A)}{1/\cos(A) + \cos(A)}$$ ...
3
votes
5answers
285 views

Verify trigonometry equation $\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$

How would I verify the following trigonometry identity? $$\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$$ My work so far is $$\frac{\sin A}{\cos A}-\frac{1}{\sin A}\frac{1}{\cos A}(1- \cos^2 A- ...
4
votes
1answer
882 views

Remembering exact sine cosine and tangent values?

There exists a common trick to remember exact sine cosine and tangent values. The trick is relatively long, so instead of reposting it, please refer to my answer on this page. Although I have used ...
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3answers
487 views

Use equalities to derive important trigonometric functions

The trigonometric functions I must know: (A) $\sin(-x)=-\sin x$ (B) $\cos(-x)=\cos x$ (C) $\cos(x+y)=\cos x\cos y-\sin y\sin x$ (D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$ $\sin^2x+\cos^2x=1$ (Use ...
2
votes
3answers
2k views

$(\sin\theta+\cos\theta)^2=1+\sin2\theta$

49) $(\sin\theta+\cos\theta)^2=1+\sin2\theta$ Left Side: \begin{align*} (\sin\theta+\cos\theta)^2=\sin^2\theta+2c\cos\theta\sin\theta+cos^2\theta=1+2\cos\theta\sin\theta \end{align*} This can ...
1
vote
2answers
80 views

Trigonometric Identities To Prove

$\tan\theta+\cot\theta=\dfrac{2}{\sin2\theta}$ Left Side: $$\begin{align*} ...