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

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0
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1answer
24 views

Finding the general formula for the solutions of the equation $\cos(2\Theta) = \frac{\sqrt{2}}{2}$

so as you can see on the title, it says finding the General Formula. So First lets take a look on the question: Solve the equation. Give the general formula for all the solutions. $$cos(2\Theta) ...
2
votes
0answers
46 views
+50

Which properties characterize $\sin, \cos$?

I know a few properties of $\sin$ and $\cos$, for example: $\sin^2+\cos^2=1$ $\sin (a+b) = \sin a\cos b+\cos a\sin b$. $\cos (a+b) = \cos a\cos b-\sin a\sin b$. $\sin (x+\delta) = \sin x$ for some ...
1
vote
1answer
16 views

Given $\tan a = -7/24$ in $2$nd quadrant and $\cot b = 3/4$ in $3$rd quadrant find $\sin (a + b)$.

Say $\tan a = -7/24$ (second quadrant) and $\cot b = 3/4$ (third quadrant), how would I find $\sin (a + b)$? I figured I could solve for the $\sin/\cos$ of $a$ & $b$, and use the add/sub ...
5
votes
4answers
82 views

Solving $2\cos\left(2\theta\right) = \sqrt{3}$

I have a question on this test review problem (that will help us on a test), and I have no clue what it's asking. We're learning trigonometry, (Analytic Trigonometry), like about the unit circle, ...
5
votes
1answer
207 views
+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
2
votes
1answer
91 views

What's the integral of $\frac{1}{x^2}\csc^2\left(\frac{1}{x}\right)$?

It's known that $\int\csc^2(x)dx = -\cot(x) + C$, but I don't know how to integrate $\int\frac{1}{x^2}\csc^2(\frac{1}{x})dx$. Can you help? Answer to integral ...
10
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0answers
125 views
+50

Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
0
votes
3answers
41 views

How to convert arccos to arctan?

Is this true? $$\arccos\frac{A}{\sqrt{A^2+B^2}}=\arctan\frac{B}{A}$$ If so, how can one show it?
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2answers
19 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
2
votes
5answers
73 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
1
vote
1answer
89 views

Don't know how to approach $x \cos(x) - 2 \cos^2(x) = 2$

I don't how to solve for $x$ because of the mixture of trig and $x$ outside a trig function. Can someone give me a hint how to proceed? $$x \cos(x) - 2 \cos^2(x) = 2$$ where the interval is $[0,2\pi]$ ...
-2
votes
0answers
17 views

To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [on hold]

Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ...
1
vote
1answer
26 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
1
vote
1answer
521 views

Drawing a fitted wave sine between any two points in 2d

I'm trying to draw a sine wave with specified start point, end point and length. To do so I have already done finding Amplitude and Period for sine function. The problem is that sine wave is not ...
2
votes
4answers
39 views

Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square.

A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ...
1
vote
3answers
45 views

Is there a trig identity to help solve this equation? $A \cdot \cos\theta = B+ \sin\theta$

I'm trying to solve for $\theta$ in a simple equation: $A \cdot \cos(\theta) = B+ \sin(\theta)$ ($A$ and $B$ are constants) But all the trig identities I've tried just make the equation worse. ...
-1
votes
0answers
24 views

PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
6
votes
4answers
169 views

How to evaluate $\lim\limits_{x\to 0} \frac{\sin x - x + x^3/6}{x^3}$

I'm unsure as to how to evaluate: $$\lim\limits_{x\to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^3}$$ The $\lim\limits_{x\to 0}$ of both the numerator and denominator equal $0$. Taking the derivative ...
1
vote
2answers
49 views

Solving $\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$

If we have to find the solutions of equation $$\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$$ Using a triangle I rewrite it as $$2 \arctan ...
1
vote
0answers
15 views

How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
178
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16answers
27k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
11
votes
6answers
281 views

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ ...
2
votes
3answers
58 views

How to show that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is strictly increasing for $x \geq 1$?

I am trying to prove that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is a strictly increasing function for $x \geq 1$. I try to do this by showing that $f'(x)>0$ for all ...
3
votes
1answer
56 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
2
votes
1answer
854 views

Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$?

Messing around with u substitution, I tried to integrate $\tan^3x$ as follows: $$ \tan^2 x \tan x = (\sec^2 x - 1)\tan x \\ u = \sec x \\ du = \sec x \tan x dx \\ du = u \tan x dx \\ du/u = \tan x dx ...
1
vote
1answer
43 views

Solve $\cot x \csc x + \cot x = 0$

$$\cot x \csc x + \cot x = 0$$ Give exact answers in radians. I tried $$\cot x(\csc x +1)=0$$ $$\cot x =0, \csc x= -1$$ But I'm not sure if that is correct. Please show me how to do this problem. I ...
2
votes
2answers
39 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
0
votes
1answer
68 views

Simple yet challenging integral, can it be solved analytically, and if so, the answer.

I'm trying to find solutions to the 3 following integrals. The first 2 are of the same form, only varying by a constant in the numerator within the cosine, and yes, x is a constant in the first one. ...
0
votes
2answers
15 views

Find unit vector that bisects two directed line segments. [on hold]

I'm trying to find the 2D unit vector that bisects two directed line segments with sign relative to the orientation of the line segments (left-hand side should be positive). Here is a graphic that ...
0
votes
1answer
29 views

Distance from Chicago to New York

An airplane flies $520$ miles from Chicago to Virginia. Then it turns $45$ degrees to face New York and flies $630$ miles to New York. What is the distance from Chicago to New York? Given the $45$ ...
6
votes
1answer
101 views

Does rationality of $\cosh(nx)$ and $\cosh((n+1)x)$ imply rationality of $\cosh(x)$?

Suppose that $x\in\mathbb{R}^+$ and $n\in \mathbb{N}$. If $\cosh(nx)$ and $\cosh((n+1)x)$ are rational, can we show that $\cosh(x)$ is rational too? I guess the following equalities should be useful: ...
2
votes
3answers
44 views

Precalculus/Trigonometric Functions of Sine, Cosine, and Tangent with given parameters?

for my precalculus class I was given an assignment for extra credit however it is some material that I have yet to cover or learn as far as sine, cosine, and tangent go. Below is the prompt that I was ...
1
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0answers
29 views

The angle giving minimum value

We know minimum value of $\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x)=6$ by AM-GM inequality. But I wanted to know whether manually can find out that angle $x$. Is it possible?
-1
votes
1answer
35 views

Find minimal possible value of the expression $4\cos^2\frac{n\pi}{9}+\sqrt[3]{7-12\cos^2\frac{n\pi}{9}},$ where $n\in\mathbb{Z}.$ [on hold]

Find minimal possible value of the expression $4\cos^2\frac{n\pi}{9}+\sqrt[3]{7-12\cos^2\frac{n\pi}{9}},$ where $n\in\mathbb{Z}.$ Is it enough to check for $\{20^\circ,\,40^\circ,\ldots,180^\circ\}$? ...
4
votes
1answer
381 views

Sum over fourth power of the sine

I am considering the sum $$ A_m = \sum_{j=0}^m \sin^4\left(\frac{j}{m}\cdot\frac{\pi}{2}\right). $$ I think that for $m>1$ it holds $$ A_m = \frac{3m+4}{8}, $$ but I can't really get to it.
2
votes
2answers
25 views

Converting specific equations from Polar to Cartesian

These different equations are given in Polar and my goal is to plot them in Cartesian coordinate system: $r = \cos(4φ)$ $ φ = \dfrac r {r-1}$, $r > 1$ I am aware of: $x = r \cos( φ )$ $y = r ...
2
votes
3answers
70 views

Prove that $\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$

Prove that $\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$ ...
0
votes
1answer
63 views

How to express $\sin \sqrt{a-ib} \sin \sqrt{a+ib}$ without imaginary unit?

I got this kind of expression as a value of an infinite product: $$\prod_{k=1}^{\infty} \left(1-\frac{A}{k^2}+\frac{B}{k^4} \right)$$ It's easy to see how it can be factored into a product of two ...
0
votes
2answers
33 views

In equilateral triangle,One vertex of a square is at the midpoint of the side, and the two adjacent vertices are on the other two sides of triangle

In the equilateral triangle $ABC,AB=12.$One vertex of a square is at the midpoint of the side $BC$, and the two adjacent vertices are on the other two sides of the triangle.Find the length of the side ...
1
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2answers
25 views

most general antiderivative involving sec x

I'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ...
3
votes
1answer
1k views

Distance between two gears surrounded by a known-length belt

This question is very similar (but not identical) to this one: Finding the distance between two gears (actually, we are trying to solve it on Bicycle Exchange: ...
0
votes
1answer
29 views

max and min sine function and all intervals

I have a calculus question: The voltage signal from a standard North American wall socket can be described by the equation V(t) = 170sin(120πt), where t is time, in seconds, and V(t) is the voltage, ...
0
votes
7answers
56 views

Show that $\sin^2 \theta \cdot \cos^2\theta = (1/8)[1 - \cos(4 \theta)]$.

I have these two problems I'm working on! First of the Double Angle Formula! This formula I attempted to do a lot but couldn't get to the identity! $$\sin^2 \theta \cdot \cos^2\theta = \tfrac18[1 - ...
0
votes
2answers
58 views

why soh cah toa is right?

i am confused by the sine of an angle, (it might appear evident for some of you but please i am not an expert ). sine of an angle is says to be the half of the magnitude of the chord of 2 time the ...
2
votes
2answers
14 views

Converting ratio of cosines to tangent or cotangent

Given the function: $f : [-\frac{\pi}{2},\frac{\pi}{2}] \to \mathbb{R}$ $f(x) = \frac{\cos{x}}{\cos{(x-a)}}$ for some $a \in \mathbb{R}$ Is it possible to convert it to some kind of translated / ...
0
votes
1answer
23 views

How to setup vector story problems

I'm studying for my trig final and I know how to do all the math, but I don't always understand how to setup the story problems. Mostly I'm struggling with vector story problems. For example: Forces ...
5
votes
1answer
157 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} ...
1
vote
0answers
27 views

3D bend equation derivation.

This is how the bend work: (The number is the angle) I was searching for an equation to bend an object in a specific axis and I found one,It worked pretty well,but unfortunately I don't know why it ...
1
vote
3answers
53 views

Prove: $\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$

This is not a homework question, its from sl loney I'm just practicing. To prove : $$\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$$ So I ...