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

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-2
votes
1answer
92 views

Find the integer solutions for $n$ in $\sin\dfrac \pi {2n} + \cos\dfrac \pi {2n} = \dfrac{\sqrt n} 2$.

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.
6
votes
5answers
380 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 $\...
0
votes
1answer
19 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 ...
3
votes
4answers
214 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
-4
votes
3answers
94 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 + \...
2
votes
0answers
32 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 ...
0
votes
1answer
30 views

Given $\iint_D \arctan \frac y x \, dx \, dy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates

Given $\iint_D \arctan \frac{y}{x} \, dx \, dy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates. I stuck with finding $\theta$. I know that $...
-1
votes
4answers
46 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)^{...
0
votes
1answer
17 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 ...
2
votes
3answers
72 views

Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
-2
votes
3answers
55 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 ...
1
vote
2answers
83 views
+200

A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
1
vote
1answer
34 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 ...
-1
votes
1answer
24 views

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

$$\ln \left(\frac{\left(1+\sqrt{5}\right)^2 \left(2+\sqrt{5}\right)}{4}\right)=\text{arcsinh }(2)+2 \text{ arccsch }(2)$$
0
votes
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 $\...
5
votes
5answers
266 views

How to integrate $\int \frac{1}{\cos(x)}\,\mathrm dx$

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 \frac{...
0
votes
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 $2x-1 \ge 0$ $\...
1
vote
1answer
84 views

why value of Trigonometric ratios of angle and its reference angle are same?

I'm learning Trigonometry right now with myself and at current about how to find trigonometry ratios of angles greater than $90^\circ$. I came to know that for finding trigonometric ratio of these ...
0
votes
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=...
0
votes
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 \...
-1
votes
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 ...
1
vote
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) ...
3
votes
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 ...
-2
votes
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? ...
1
vote
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\...
4
votes
1answer
190 views

Geometry and chemistry

There's a really obvious geometric reason why the cosine of the bond angle in graphite is $-1/2$: the stuff consists of sheets shaped like honeycombs. There's also a really obvious geometric reason ...
4
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
1answer
393 views

How do you find the radius of an arc given arc length and height?

. Please excuse the poor drawing, but how would you go about solving this problem? Known: Arc length Height (I'm not sure what the proper term for this parameter is.) Unknown: Sagitta Chord ...
0
votes
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 ...
0
votes
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 $7.51\tan{\theta} - 2.656(\tan{\theta})^2 - 2.656=0$ ...
22
votes
2answers
640 views

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

Show that $$\sum_{n=1}^{\infty}\frac{\sinh\big(\pi n\sqrt2\big)-\sin\big(\pi n\sqrt2\big)}{n^3\Big({\cosh\big(\pi n\sqrt2}\big)-\cos\big(\pi n\sqrt2\big)\Big)}=\frac{\pi^3}{18\sqrt2}$$ I have no ...
2
votes
3answers
393 views

Trigonometry Problem Solving

How can we estimate the height (h) of a castle surrounded by a moat, using the info below?
1
vote
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 ...
0
votes
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 ...
4
votes
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 ...
2
votes
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 + \...
2
votes
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?
1
vote
1answer
393 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
6
votes
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 ...
0
votes
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)...
63
votes
9answers
1k views

Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
0
votes
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 ...
2
votes
3answers
79 views

Find maximum $2\sin 5x-3\cos x$.

Is it possible to find the maximum of $2\sin 5x-3\cos x $ without using calculus nor numerical methods? I suspect there is a way to play around with trig identities until the expression is only in ...
2
votes
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 ...
1
vote
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(\...
2
votes
2answers
198 views
+50

The relationship between tan(x) and square roots

Please note: I am working in DEGREES :) I think the easiest way to illustrate my point is by showing some examples: $ \tan(0) = \sqrt 0 = 0$ $ \tan(22.5) = \sqrt 2 -1$ $ 3 \cdot \tan(30 ^\circ) =\...
0
votes
5answers
2k views

Harder Trigonometry Identity ($\sec A+\csc A$)

How do I prove: $\sin A (1 + \tan A) + \cos A (1 + \cot A) = \sec A + \csc A$ I've tried expanding the brackets by multiplying sin A and cos A to the left hand side but to no avail. Where should I ...
0
votes
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 ...