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

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6answers
12k views

$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?

How does that bit work? How is $$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
2
votes
2answers
173 views

How to get angle bewteen two vectors in range -1 to 1 without using arc cosine?

Given two normalized vectors in 3d space, how can I get a value from $-1$ to $1$ based on their angle without using arc cosine? With use of arc cosine, I think this would give me the correct result. ...
1
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0answers
130 views

Shortest time taken for a targeted object with a set speed to meet a body orbiting in a circle

I'm trying to figure out how to find the optimum point that a ship in 2D space would meet a planet which was orbiting in a perfect circle. The orbit is at a constant rate, and the ship can only move ...
1
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2answers
121 views

Proving that $\sin(a)\cos(b)$ and $\cos(a)\sin(b)$ identities are identical using $\sin(-x)=-\sin(x)$

This website states the two trig identities below are identical: [\begin{array}{l} \sin (a)\cos(b) = \frac{1}{2}\left[ {\sin (a + b) + \sin (a - b)} \right] \Rightarrow 1\\ \cos (a)\sin (b) = ...
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0answers
520 views

Solving the equation $\cos(x) \cdot \cosh (x) + 1 = 0$

$$\cos(x) \cdot \cosh (x) + 1 = 0$$ Sorry I am a software developer and I have forgotten this part of mathematics! What is the value of $x$ in the above equation? I need the steps to solve the ...
7
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2answers
131 views

Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$

Show that $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
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1answer
290 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
3
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1answer
68 views

$\tan B\cdot \frac{BM}{MA}+\tan C\cdot \frac{CN}{NA}=\tan A. $

Let $\triangle ABC$ be a triangle and $H$ be the orthocenter of the triangle. If $M\in AB$ and $N \in AC$ such that $M,N,H$ are collinear prove that : $$\tan B\cdot \frac{BM}{MA}+\tan C\cdot ...
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3answers
281 views

Can't find solutions for $\tan{2x} = \tan{x}$

Solving $\tan{2x}=\tan{x}$ Reducing the left side: $$\frac{\sin{2x}}{\cos{2x}} = \frac{2\sin{x}\cos{x}}{2\cos^{2}(x)-1}.$$ Reducing the right side: $$\tan{x} = \frac{\sin{x}}{\cos{x}}.$$ therefore: ...
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2answers
117 views

$\frac{AB}{A'B'}+\frac{BC}{B'C'}+\frac{CA}{C'A'} \geq 4 \left(\sin{\frac{A}{2}}+\sin{\frac{B}{2}}+\sin{\frac{C}{2}}\right). $

Let be a circle inscribed in the triangle $\triangle ABC$ wiht the center $I$. The intersection of the circle with $AI$ is $A'$, with $BI$ is $B'$ and with $CI$ is $C'$. Prove that: ...
3
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2answers
91 views

$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $

Help me please to prove that: for any $\triangle ABC$ we have the following inequality: $$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $$ It's about ...
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2answers
328 views

How to count angle between vector and horizontally oriented vector?

I need to calculate in my Java application an angle between my line and horizontal line that has the same beginning. I have a line described by its equation: $$f(x) = ax + b.$$ I would like to know ...
3
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2answers
508 views

How to get the minimum angle between two crossing lines?

I'm not a student, I'm just a programmer trying to solve a problem ... I just need the practical way to calculate the smallest angle between two lines that intersect. The value, of course, must always ...
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3answers
341 views

Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$

Or put differently, does $$\lim_{n \to \infty}\big(\max \{\sin 1, \sin 2, \ldots ,\sin n\}\big) = 1?$$ My intuition says yes, but how can one prove this?
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1answer
65 views

Trigonometric Sums - URSS

Calculate the value of the sums: (a) $\cos x+\binom{n}{1}\cos 2x +\cdots+\binom{n}{n} \cos (n+1)x $; (b) $\sin x+\binom{n}{1}\sin 2x +\cdots+\binom{n}{n} \sin (n+1)x $.
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1answer
48 views

approximation of law sines from spherical case to planar case

we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
1
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3answers
170 views

Why does tan(t) touch the unit circle at (1,0)?

I can't get my head around this, any help would be very much appreciated. Thanks EDIT: t is an angle, where 0 < t < 90, angle t is in degrees EDIT: Added a picture I lifted from google
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3answers
212 views

Show that for all $(x,y)$ there exists $(r,\theta)$…

Given a problem wherein $(x,y) \in \mathbb{R}^2$, I often transform to polar coordinates by introducing the assumption that $(r,\theta)$ satisfy $x = r\cos \theta, y = r\sin \theta.$ Of course, it's ...
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2answers
337 views

A question on Trigonometry (bisector)

If two bisector of a triangular is equal, then it is Isosceles triangular.
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1answer
88 views

The law of cosines for a sphere

$\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(C)$ Prove that if $a$, $b$, and $c$ is approximately $0$, then $c^2 = a^2 + b^2 - 2ab~\cos(C)$. I wasn't sure how to prove this. One thought I had was ...
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5answers
3k views

In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?

I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason why $\tan ...
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0answers
127 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
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2answers
164 views

trigonometry equations

Take this question: "We follow the tips of the hands of an old fashioned analog clock (360 degrees is 12 hours) . We take the clock and put it into an axis system. The origin (0,0) of the axis ...
4
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1answer
140 views

finding $\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$

I want to find $$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$ I already know that I need to split the expression $\frac{2r}{1-r^2+r^4}$ of the form ...
1
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1answer
53 views

Uncertainty in measurements: if $x$ has uncertainty $\pm\epsilon$, what is the uncertainty in $\sin x$?

I have two questions regarding uncertainties in measurements. First, if I have some measured value for $x$ with an uncertainty $\pm e$, what would be the uncertainty in $\sin x$, $\pm\sin e$? ...
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5answers
540 views

Complex numbers and trig identities. I've heard this question is easy but I don't know how. Help?

Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make... $$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)= ...
3
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5answers
262 views

elegant proof that $\sin(x)\cdot\cos(x)=\sin(2x)/2$

I tried for a few days to prove the identity $\sin(x)\cos(x)=\frac{\sin(2x)}{2}$ and finally got the following proof. I wanted to know if someone knew a simpler or more elegant way to proof it. ...
2
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1answer
36 views

Missing rule for this simple identity?

Is there a simple rule I am missing to help prove the identity below? Of course I can find the common denominator, do multiplication, summarize, apply trigonometric product-to-sum identities, and then ...
4
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1answer
3k views

Where are secant, cosecant, and cotangent applied?

The textbook we use says: The cosecant function and the secant function are the reciprocal functions of the sine function and the cosine function, respectively, and thus are also periodic ...
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1answer
3k views

Solving the inverse of cos^2

The following equation provides the inclination ($i$) of a galaxy, using the ratio of its two axes: $$ \cos^2 i = {(b/a)^2 − (b/a)^2_{eos} \over 1 − (b/a)^2_{eos}} $$ All I need however is to ...
3
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3answers
325 views

Find minimum of $\sin^4 x + \cos^4 x + \sec^4 x$

I tried manipulating the terms but I couldn't get anywhere. The only other thing I can observe is that the minimum must be greater than $1$ since all the terms are non-negative and the range of ...
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2answers
129 views

Can you express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos?

I was going through some of my notes and found they said we can express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos. However, I ...
3
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3answers
75 views

Trigonometric Functions And Identities Question .

If $A$ , $B$ and $C$ are the angles of a triangle , I have to show that : $ \tan^2 \cfrac{A}{2} + \tan^2 \cfrac{B}{2} + \tan^2 \cfrac{C}{2} \ge 1 $ . I had only arrived at $A+B+C = \pi $ , thus ...
3
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3answers
3k views

What are some useful tricks/shortcuts for verifying trigonometric identities?

What "tricks" are there that could help verify trigonometric identities? For example one is: $$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$
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2answers
75 views

Area of a rectangular triangle

We need to calculate the area of the triangle shown in figure: The text of the problem also says that: $\sin \alpha =2 \sin \beta$. What is the area of ​​the triangle?
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2answers
35 views

For the angles a and b it is given that:-

For the angles $a$ and $b$ it is given that: $\sin a = 0.6$, $\sin b = 0.8$, $\cos a = 0.8$ and $\cos b = 0.6$ Find the value of $\sin( a + b)$ with out calculator :- thanks all - happy mother day ...
2
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1answer
512 views

Finding side and angle of isosceles triangle inside two circles

I'm having a problem that I'm not sure how to solve (or if it's even possible). It's not homework, just something I'm struggling with for a project. :) Basically, there are two circles, represented ...
2
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1answer
430 views

A controlled trapezoid transformation with perspective projecton

I'm trying to implement a controlled trapezoid transformation in Adobe Flash's ActionScript using the built-in perspective projection facility. To give you an idea of how the effect looks like: ...
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2answers
124 views

Trignometry - Cosine Formulae

c = 21 b = 19 B= $65^o$ solve A with cosine formulae $a^2+21^2-19^2=2a(21)cos65^o$ yield an simple quadratic equation in variable a but, $\Delta=(-2(21)cos65^o)^2-4(21^2-19^2) < 0$ implies the ...
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7answers
608 views

Proving $\sin (x)=\cos (90^\circ-x)$

I'm interested in the different ways of proving this, any proof is welcome. I understand one way is the cosine sum/difference formula, another is using a right angled triangle. Are there any others? ...
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2answers
571 views

Trigonometric Identities - Assignment

How do I simplify Cos(5 theta)? I got as far as Cos(2theta + 3theta). Do I then say Cos(2theta + 3theta) = Cos(2theta) + Cos(3theta)? In that case, how do I get Cos(3theta)?
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4answers
91 views

$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$

I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can ...
0
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1answer
52 views

Solving for a variable

Assuming that we're given the function: $$I_r(n) = \cos^{n-1}(\dfrac{45°}{n-1})$$ Which models the remaining intensity of a wave of light. n is how many polarizers must be placed (i.e. more ...
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5answers
83 views

Trigonometry help please? [closed]

Show that $\cos(A-B)-\cos(A+B)=2\sin A\sin B.$ Using compound angle and identity for $\cos.$ Anybody have an idea how to solve this question? Cheers all...
3
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7answers
926 views

How do I show $\sin^2(x+y)−\sin^2(x−y)≡\sin(2x)\sin(2y)$?

I really don't know where I'm going wrong, I use the sum to product formula but always end up far from $\sin(2x)\sin(2y)$. Any help is appreciated, thanks.
1
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0answers
213 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
0
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2answers
137 views

How to find the trigonometric identities?

We find many sources on the internet that provides a table of trigonometric identities (like http://www.fis.ufba.br/~luciano.abreu/TABELA.pdf), but I'd like to know if there is a way to determine ...
12
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3answers
880 views

Why does rational trigonometry not work over a field of Characteristic 2?

I am interested in solving triangles in a finite field with a computer program. Rational trigonometry seems well suited to do this. However, the Wikipedia article, as well as several published ...
2
votes
2answers
1k views

Express trigonometric expressions in terms of one trigonometric function

What is the general process to solve problems such as this: I'm preparing for this type of exam problem. For Reference, similar/more advanced problem (with solution): http://prntscr.com/ws6xl
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0answers
75 views

How were trigonometrical functions and its inverses discovered?

Imagine you just did a circle. Some functions are just definitions, like $\sin$,$\cos$ and $\tan$ but how do you derive a formula to get the $\sin$ from an angle in radians (maybe by Taylor series, ...