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
851 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 ...
0
votes
2answers
32 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
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2answers
38 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 ...
3
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0answers
26 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 ...
0
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1answer
56 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. ...
2
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2answers
45 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 ...
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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
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1answer
23 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
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1answer
95 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
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3answers
39 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 ...
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0answers
27 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?
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1answer
33 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
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1answer
371 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.
10
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4answers
219 views
+100

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 $2\pi$ ...
0
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1answer
47 views

A difficult trigonometric equation

Let $0\lt a, b\lt 1$ be two constants. Then, how can I solve the trig equation $$\{1-a\cos(\theta-\alpha)\}\{1-b\cos(\theta-\beta)\}=\dfrac14(1-a^2)(1-b^2)$$ for $\theta$ in terms of $a, b$ and ...
2
votes
2answers
24 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
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3answers
68 views

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

Prove that $\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$ ...
0
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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 ...
7
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0answers
67 views

Is there a closed form formula for $\sum_{k=2}^n\left(\frac{\sin x}{\sin\frac{x}{k}}\right)^2$ or $\sum_{k=2}^n(\csc\frac{x}{k})^2$?

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
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2answers
31 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 ...
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2answers
23 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
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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
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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, ...
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7answers
53 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 - ...
3
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0answers
34 views

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 ...
0
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2answers
56 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
13 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
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1answer
22 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
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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} ...
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0answers
19 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 ...
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3answers
51 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 ...
2
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1answer
35 views

Find smallest k for which the inequality holds

The smallest positive number $K$ for which the inequality $|\sin^2 x - \sin^2 y| \le K|x-y|$ holds for all $x,y$ is
4
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0answers
194 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 ...
0
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4answers
52 views

Why is cos -x equal to cos x?

Can somebody explain to me graphically or intuitively why this is true? Because if we think in terms of the base of the triangle formed inside a unit circle it seems like it should not be negative.
0
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1answer
20 views

Find point on rectangle where vector intercepts [on hold]

I have a vector in the centre of a rectangle pointing out of the rectangle. The size of the rectangle is known. The vector is known. The magnitude of the vector is always greater than the distance to ...
5
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4answers
217 views

A String Tied Around The Earth

Say you're standing on the equator and you have a string below you tied around the equator (40,075 km) that is the length of the equator + 1 meter (40,075.001 km). What is the maximum height you can ...
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1answer
30 views

How to correctly find CSC of an angle?

Alright, so one of my questions is CSC (angle) -5. When I plug CSC in my calculator, it says "math error." I'm using a Casio fx-300 MS, and using shift + cos, then putting an angle, such as 90.
0
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1answer
456 views

Calculation of Nutation and Rotation from Pitch and roll (yaw is fixed to 0)

i am stuck attempting to convert two angles (pitch and roll) to represent tilt within a circle. Take a plane $(Z\ \text{(Vertical)}, X\ \text{(Roll)}, Y\ \text{(Pitch)})$ If i have $45$ degrees ...
3
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2answers
176 views

Write the complex number in trigonometric form (homework question)

Write the complex number in trigonometric form, once using degrees and once using radians. Begin by sketching the graph to help find the argument θ. (Do not use cis form.) $$−1 + i$$ My work: I ...
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0answers
31 views

Possible to isolate x for: y = x + sin(x)

I just recently learned that: $x = \frac{b}{a + 1}$ when $ ax + x = b$. This got me thinking about trig functions in a similar format, i.e. can we isolate $x$ for: $y = x + sin(x)$?
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1answer
11 views

How do the coefficients in the linear combination of cosines impact the number of local minima of the sum?

Consider the following function: $$f(\theta) = r_0 + r_1 \cos(\theta + \phi_1) + r_2 \cos(2\theta + \phi_2)$$ where $\theta$ is an angle between 0 and $2\pi$. For all $0\leq k\leq 2$ we have $r_k\geq ...
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2answers
53 views

Periodic functions $f(x)=\sin{x}+\cos{x}$ [on hold]

Find period of functions $$f(x)=\sin{x}+\cos{x}$$and $$f(x)=|\sin{x}|$$ I need some hints and directions.
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3answers
50 views

A simple geometrical question regarding three circles and a line. Trigonometric construction. [on hold]

In Figure 1 three tangential circles all have the radius of 1 or r. What is the ratio of the blue line to the yellow line in terms of r, and in terms of r=1?
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2answers
1k views

How is the radian measure of angles derived/defined?

I'm currently studying angles and their measures and I've just been told that $\pi$ is the ratio of a circle's circumference to its diameter, so: $\pi =\dfrac {C}{d}$ $\pi =\dfrac {C}{2r}$ $ 2\pi ...
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1answer
57 views

Does there exists an addition formula for $\arctan(2a)$? [closed]

I want to find an expression for $\arctan(2a)$ in terms of a sum of $\arctan$s, so $\arctan(2a) = c_1\cdot \arctan(b_1)+ c_2\cdot \arctan(b_2)$. Does there exists such a formula? I tried to derive it ...
1
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2answers
21 views

Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
0
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1answer
53 views

is there an existing formula in finding the area of a rhombus wherein only the side is given?

is there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given.
7
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3answers
1k views

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$

I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me: Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers ...
1
vote
1answer
50 views

How can I compute this integral $\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $

How can I compute this integral $$\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $$ Even when I use Maxima, it does not give result. Thank you very much.
0
votes
2answers
39 views

Polar equation of the curve y = sinx

I am looking for the polar equation of the following curve given in Cartesian Coordinates. y = sinx Any kind of hint or help is appreciated.