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

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1answer
13 views

proving that triangles $ABC$, $A'B'C'$ are congruence

Given $AD$ is a median to $BC$ in triangle $ABC$, and $A'D'$ is a median to $B'C'$ in triangle $A'B'C'$, and $AD=A'D', AC=A'C', AB=A'B'$. How can i prove that triangles $ABC$, $A'B'C'$ are congruence?...
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1answer
13 views

Equation of a circle in polar coordinates under a linear transformation

Let's say we translate a circle with origin $(0,0)$ on the x axis by some constant $c$. What would the new equation of the circle be in polar coordinates? I have tried subbing in the equation of the ...
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0answers
6 views

Nonlinear odd real sinusoidal functions

I need a class of odd nonlinear sinusoidal functions whose graphs are given here: I got some example functions: 1) $x = \cfrac{x_{\max}}{2}\times\sin(\cfrac{\pi y}{y_{\max}})$ where $x_{max}$ and $...
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0answers
19 views

Alternative derivation of Euler's product formula for sine

Euler's product formula states that: $$\sin(x)=x\prod_{n=1}^{\infty}\left[1-\frac{x^2}{\pi^2n^2} \right].$$ There is also a very simple formula for another product representation for the sine ...
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1answer
25 views

Proving $\cos(a+b)=\cos a\cdot\cos b - \sin a\cdot\sin b$ [on hold]

I would like to prove the following $$\cos(a+b)=\cos a\cdot\cos b - \sin a\cdot\sin b.$$
2
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1answer
18 views

Deduce the relation from the given trigonometric relation

If $$\frac{\tan3A}{\tan A}=k$$ Then prove that $$\frac{\sin3A}{\sin A} = \frac{2k}{k-1}$$ I tried this, $$ \tan3A = \frac{3\tan A-\tan^3 A}{1-3\tan^2 A}$$ then divided by tan A on both sides and ...
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3answers
34 views

prove the following relation,

If $$ xy + yz + zx = 1 $$ then show that, $$\frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} = \frac{4xyz}{(1-x^2)(1-y^2)(1-z^2)}$$ I have tried multiplying all three terms on the left side, and ...
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2answers
30 views

How to calculate A in sin Ax if sin x = sin Ax? [on hold]

if sin x = sin ax, is a 180/pi or pi/270, or 270/pi, or pi/180. How do I calculate the value of a? Arun
2
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0answers
33 views

Names for related pairs of angles

I seek the names (if they exist) of two relationships between angles. Two angles are complements of each other if they add up to a quarter circle. $\sin\alpha=\cos\beta$ and vice versa. Two angles ...
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1answer
21 views

Value of a product of cosines and the floor of its reciprocal

$$ \mbox{The question states}\quad {a \over b} =\prod_{n = 1 \atop{\vphantom{\LARGE A}n \not= 9}}^{17}\cos\left(n\pi \over 18\right) $$ $$\mbox{And it is also provided that}\quad \left\lfloor{b \over ...
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1answer
21 views

Overlapping area of two circle's crossing it's center i.e., length of overlapping is greater than r of the circle. Circle's has equal area.

Let there be two circular coasters of equal area (and negligible height). The purpose of is to find how far the two coasters need to be moved on top of each other such that the area of the overlapping ...
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4answers
48 views

Limit to infinity of trigonometry

\begin{align*}\lim_{n\rightarrow \infty}\frac{n\left(\left(1-\cos^2\frac{16}{n}\right)\sin\frac{16}{n}\right)^{1/3}}{4}=\lim_{n\rightarrow\infty}\frac{n\left(\sin^2\frac{16}{n}\sin\frac{16}{n}\right)^{...
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2answers
124 views

Find the integer solutions of $\sin\frac \pi {2n} + \cos\frac \pi {2n} = \frac{\sqrt n} 2$ [on hold]

Let $n$ be a fixed positive integer such that $$\sin\dfrac \pi {2n} + \cos\dfrac \pi {2n} = \dfrac{\sqrt n} 2$$ then find the value of $n$. I have no clue how to do this sum. I couldn't even try it.
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3answers
57 views

How to maximize $\cos\theta$?

I have a question about maximizing $\cos\theta$. I have the equation $y=H\cos\theta$, where $H$ is the fixed height of a triangle. The problem asks me to maximize $\cos\theta$, but I have no idea ...
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5answers
388 views

Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$

Find the value of $$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$ My attempt I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\...
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1answer
27 views

How to prove such a hyperbolic sine cosine related equality? [on hold]

$$\ln \left(\frac{\left(1+\sqrt{5}\right)^2 \left(2+\sqrt{5}\right)}{4}\right)=\text{arcsinh }(2)+2 \text{ arccsch }(2)$$
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1answer
35 views

Can Someone help me with my trigonometry rotation, formula? [on hold]

I've been working on some code for a game to make a hit box, this question is just about the math though. Basically I'm trying to rotate an X, Y point(i guess according to the game it's Z,X Not sure ...
0
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1answer
49 views

Help simplifying $\sum_{n=0}^\infty \cos(n\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$

In a proof of $\sum_{n=0}^\infty \cos(n\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$ I need help figuring out the identity used to simplify from red $ \color{red}{1}$ to $\...
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1answer
23 views

Inverse Trigonometric piece-wise functions

I was solving the equation $$2\tan^{-1}(2x-1)=\cos^{-1}x$$ Now while solving the question, the author of the book has written only the first case in the solutions manual. CASE I $2x-1 \ge 0$ $\...
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0answers
38 views

Maximising sum of sine/cosine functions

I have got a problem and I would appreciate if one could help. I have to maximise following function that is the sum of sine/cosine functions: $$ f(x,y)=a_1 \cos(x) +b_1 \sin(x)+ a_2 \cos(y) +b_2 \...
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0answers
22 views

Trig funct graphs check (amplitude, period)

Hi for the following questions I was wondering if I was correct in my answers and if I am incorrect, please correct me. Thank You My solutions, please correct me if I am wrong. 2cosx, amplitude ...
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0answers
14 views

Start and endpoint of line, creating arrow heads [on hold]

I have a start point(5.6,4) and an endpoint (6.1,3.15) I want to make an arrow head at the start point that is an equilateral triangle(60 degrees) with a length of .1. How can I accomplish this? ...
3
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2answers
48 views

Simplifying trig expression $\frac{1}{1-\cos \theta}$

I need help with the following trig problem, I'm getting the first part, but can't seem to complete it. $$\frac{\cos \theta}{1-\cos^2 \theta}- \frac{1}{1-\cos \theta}$$ The first part is going to ...
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3answers
79 views

Find $f'(x)$in terms of $f(x)=|\cos(x)|\sqrt{1-\cos(x)}$

I am trying to solve the following exercise : Let $f$ be the function defined by : $$\forall x\in]0,\pi[\;\;\;\;\; f(x)=|\cos(x)|\sqrt{1-\cos(x)}$$ calculate $f '(x)$ in terms of $f(x),$ for all $x\...
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1answer
39 views

Doubt regarding signs in trigonometry equations

I have been trying to solve some equations, and for the same I found an online answer. Here's the link - http://citeseerx.ist.psu.edu/viewdoc/downloaddoi=10.1.1.456.6096&rep=rep1&type=pdf#page=...
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1answer
15 views

Plotting triangles based on a single point with distance and angle.

I'm tasked with creating an arrowhead within a pdf program. I have a single point with at $x=5.6$, $y=4$ this would be point A of my triangle I want to make the sides equal at $90$ degrees angles ...
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3answers
60 views

$\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get.

$\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get. Here $0<\alpha < \frac{\pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$. Yes I ...
4
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1answer
44 views

Maximum and minimum of $f(x)=\cos(\sin(x))-\sin(\cos(x))$

Given the function: $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ it has absolute maxima at $x=(2k+1)\pi$ with $k=0,1,..N$ and relative maxima at $x=2k\pi$. It is not clear where are the minima. Putting the ...
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0answers
10 views

Length of elliptical segment given starting and ending points and slope

I would like to represent the flight path of a turning aircraft with an ellipse. Initially, the baseline turn is 180 deg, with a constant radius. The speed of the aircraft is constant. During the ...
1
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1answer
32 views

Angles of lines tangential to a circle

I am looking to find the angles of line features relative to the tangent of a circle. Please see this example for general idea. Angles to line features (purple) I am looking for are (poorly drawn) ...
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1answer
24 views

How to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$

I'm trying to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$ and the way that it's been done in my notes is by somehow changing the equation to $7.51\tan{\theta} - 2.656(\tan{\theta})^2 - 2.656=0$ ...
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0answers
22 views

What maths would most likely have used for this game's horizontal bullet spread? Firing at 90° y causes the marks to line up perfectly.

While playing Doom, a game with a lot of mathematical techniques for various things, if I aim my x-as-well-as-y-spreading shotgun up at a 90° on the y view angle (x and y angles are used to look ...
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3answers
49 views

Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
4
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6answers
83 views

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$ The easiest way was to just look at the graph and I found out that the region is $x \in ({1\over \sqrt{2}} , 1]$ But I ...
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2answers
39 views

How to prove that a sum of $\cosh(kx)$ is equal to a formula? [duplicate]

I need to prove that $$\sum_{k=0}^{n}\cosh(kx) = \frac{\sinh((n+1/2)x) + \sinh(x/2)}{2\sinh(x/2)}$$ Can you help me out? How do I even start?
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1answer
28 views

Finding $f(x)$ in $\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$

I need to find a valid $f(x)$ such that: $$\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$$ I can apply the FToC and I get: $$(2\cos(x)-\sin(x)f(x))+(\cos^2 x f'(x))=2x\sin(x)\cos(x)...
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7answers
153 views

If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$

$\sin x + \sin y = 1$ $\cos x + \cos y = 0$ Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured. I got the question from chapter 26 of a comic ...
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0answers
35 views

Math precalculus/trig

Circle $O$ below has radius 1. Eight segment lengths are labeled with lowercase letters. Six of these equal a trigonometric function of $\theta$. Your answer to this problem should be a six letter ...
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1answer
44 views

What is the problem in my computation of $\sin 18^{\circ}$?

I needed to compute $\sin 18^{\circ}$. Now, these two relations hold for every $x$: $\cos 5x=16\cos^5x-20\cos^3x+5\cos x$ $\sin5x=16\sin^5x-20\sin^3x+5\sin x$, which can be easily proved using the ...
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5answers
78 views

Prove the following trigonometric result

If $\theta_1,\theta_2(0\leq\theta_1,\theta_2<2\pi)$ are two solutions of $\sin(\theta+\phi)=\frac{1}{2}\sin(2\phi)$, prove that $$\frac{\sin(\theta_1)+ \sin(\theta_2) }{ \cos(\theta_1)+ \cos(\...
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1answer
42 views

Changing the period of sine versus arc length

Let's consider $ y = \sin x $. Let $ s \in \mathbb{Q} $ and $ s > 1 $. One may calculate the arc length of sine between $ 0 $ and $ 2\pi s$ using the formula: $$ L = \int_0^{2\pi s} \sqrt{1 + \...
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3answers
98 views

Find the value of $6P_{10} - 15P_8 + 10P_6+7$ for $P_n=\sin^n x+\cos^n x$

If $P_n=\sin^n x+\cos^n x$ where $n$ is a whole number and $x$ is a real number. Find the value of $6P_{10} - 15P_8 + 10P_6+7$ I tried this: $$P_6 \Longrightarrow \sin^6 x + \cos^6 x = (\sin^2 x + \...
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4answers
68 views

find minimum value of $2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$

find minimum value of $2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$ I have found the minimum value using derivative method : Let $f(\theta)=2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$. Then calculate $f'(\...
1
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1answer
38 views

Find $\tan 2x$, given $\tan(x+y)=3$ and $\tan(x-y)=2$

I am having a hard time to solve this trigonometric system of equations. The equations is as follows: We are given $$\tan(x+y)=3$$ $$\tan(x-y) = 2$$ and we need to find $$\tan2x$$ I have ...
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1answer
14 views

Bearings question confusion

At 12.00pm , a ship was spotted at a point P , 30 km due north of an island , L . The ship was sailing on a bearing of 120 degree at 32km/h . How far was the ship from the island at 12.30pm ? My ...
0
votes
1answer
50 views

Why don't we take $\sin x$ as negative square root of $1-\cos^2x$? [on hold]

I am confused of using $\sin x$ as as negative square root of $1-\cos^2x$. Can anyone help please?
1
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1answer
18 views

Sides of a triangle are in Arithmetic Progression, then find $\tan (\alpha+ \frac{\beta}{2})$

The sides of a triangle are in Arithmetic Progression. If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds the smallest angle by $\beta$, then find the value of ...
3
votes
3answers
53 views

How to find $ \tan \left(\frac{x}{2}\right) $ knowing that $\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $

Good evening to everyone. I don't know how to find $ \tan \left(\frac{x}{2}\right) $ knowing that $$\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $$ and x$\in (0,\frac{\pi}{3})$ Here's what I've ...
0
votes
0answers
17 views

Hypocycloid with an outer ellipse

I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, ...
0
votes
0answers
15 views

Cycloids with ellipse

I have been researching about the epitrochoids and hypotrocoids lately. I was wondering if it would be possible for us to have an ellipse rolling around a circle? If so, then how could one derive its ...