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
12 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
2
votes
2answers
53 views

If $ \tan(20^{\circ}) = p $, find $ \frac{\tan(160^{\circ}) - \tan(110^{\circ})}{1 + \tan(160^{\circ}) \tan(110^{\circ})} $.

I applied the $ \tan(A - B) $-formula to make it $\tan(50^{\circ}) $, then I split it to $ \tan(30^{\circ} + 20^{\circ}) $. My answer came out to be $ \dfrac{\sqrt[3]{p + 1}}{\sqrt[3]{- p}} $, but ...
31
votes
3answers
652 views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...
1
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0answers
22 views

Realistic Bounce (Using Trig?)

background: I am making a graphics program where the major purpose of it is to have a ball (traveling on an arbitrary slope) to bounce realistically off of a line (which is also at a arbitrary slope). ...
3
votes
1answer
32 views

arccos and arcsin integral contradiction:

I am shown: $$f(x) = \arcsin x \implies f'(x) = \frac{1}{\sqrt{1-x^2}}$$ $$f(x) = \arccos x \implies f'(x) = -\frac{1}{\sqrt{1-x^2}}$$ These two derivatives can be very readily derived by a bit of ...
3
votes
1answer
71 views

Finding $\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$

Finding $$\mu=\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$$ I thought $$z^n=1=e^{i2\pi}\implies z=\cos\frac{2k\pi}n+i\sin\frac{2k\pi}n\quad k\in\{1,2,...,n-1\}$$ Now we have: ...
0
votes
1answer
62 views

Find largest possible value of $x+y$

If $4\sin x. \cos y + 2\sin x+2\cos y+1=0$, find the largest possible value of the sum $(x+y)$. How do I manipulate my expression? I am not getting $(x+y)$ form. Thanks.
4
votes
4answers
91 views

Evaluate $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$ using a double integral

I was given the following problem: Evaluate the following integrate using a double integral: $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$. The professor told us off the bat the answer was ...
1
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0answers
29 views

Converting solutions to separation constant to Cosh and Sinh

The Laplace's equation inside a rectangle is: $$u_{\text{xx}}+u_{\text{yy}}\text{=0}$$ The IC's are: $${u(0,y)=g(y)}$$ $${u(L,y)=0}$$ $${u(x,0)=0}$$ $${u(x,H)=0}$$ Via method of separation we have ...
1
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1answer
35 views

Limit of functions involving trigonometry as n approaches infinity

By graphing these functions, I know that P(n) approaches pi as n tends towards infinity. However, is there a mathematical way for proving this? I am doing a maths exploration on Archimedes' ...
0
votes
1answer
32 views

Is there an algorithm to determine if an arc through 3 points is concave up or concave down?

Armed with only the three points in 2-dimensional space, $X = \{x_1, x_2, x_3\}$, is there a simple inequality or algorithm that can return whether or not an arc $A$ through these three points is ...
3
votes
3answers
32 views

Equivalence of Solutions to Wave Equation

The differential equation $$\ddot x = -\omega^2 x$$ apparently has solutions of $$x = Ae^{i\omega t} + Be^{-i\omega t} \tag{1}$$ AND $$x = A\sin(\omega t) + B\cos(\omega t) \tag{2}$$ AND $$x = ...
0
votes
1answer
22 views

Smooth function, lateral limit and trigonometry

Let $\theta: (t_0-\varepsilon, t_0)\to\mathbb{R},\ \theta\in C^{\infty}((t_0-\varepsilon,t_0))$. Knowing that the following two limits exist: $\lim\limits_{t\nearrow t_0} \cos^2 \theta(t)$ and ...
6
votes
6answers
74 views

System of equations involving sin and cos

I'm trying to solve the following system: $$ \sin(x) + \cos(y) = 0.6\\ \cos(x) - \sin(y) = 0.2\\ $$ Solving for y in terms of x: $$ y=\arccos(0.6-\sin(x))=\arcsin(\cos(x) -0.2) $$ Therefore: $$ ...
3
votes
5answers
151 views

Trigonometric Functions Limit

In my assignment I have to solve the following question. I know the answer, but I keep getting it wrong, and I don't know how to solve it. $$\lim_{x \to 0} \frac{1-\cos x}{x\sin x}$$ I have tried ...
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3answers
18 views

Find the values of $x$ satisfying $\sin^{-1}(|\sin x|)-\cos^{-1}(\cos x)\ge0$ in $[0, 2\pi]$

Find the values of $x$ satisfying $\sin^{-1}(|\sin x|)-\cos^{-1}(\cos x)\ge0$ in $[0, 2\pi]$. I think it would be better explained by drawing the graphs. Kindly help me in this question.
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2answers
43 views

The number of pairs $(x,y)$ of real numbers satisfying $|\tan(\pi y)|+ \sin^2(\pi x)=0$ and $x^2 + y^2\le2$ [closed]

Here I have a question: Find the number of pairs $(x,y)$ of real numbers satisfying the following: $$|\tan(\pi y)|+ \sin^2(\pi x)=0\quad\textrm{and}\quad x^2 + y^2\le 2$$ The answer is ...
0
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1answer
18 views

Converting cos(x-30) to two terms

I have a worked example in front of me for a particle kinetics and kinematics question. In the working it has a an equation: $$0.6=\frac{V_0\sin(60)+V^`cos(x-30)}{10cos(30)}$$ And on the next line ...
0
votes
1answer
22 views

Find sides of triangle

I already know two sides $a_2$, $b_2$ and the angle $C$. I don't know angles $A$, $B$ and sides $a_1$, $b_1$, $c_1$, $c_2$. How can I find $a_1$, $b_1$, $c_1$? Or How can I find $c_2$? Here, ...
-1
votes
0answers
41 views

Write the word or phrase that best completes the statement or answers the question. [duplicate]

Write the word or phrase that best completes the statement or answers the question. $$\cos^2\theta-\sin^2\theta=1+\sin\theta$$ i have tried manipulating the equation by substituting $x^2$ and $y^2$ ...
2
votes
1answer
62 views

$\cos^2(\theta)-\sin^2(\theta)=1+\sin(\theta)$ over the interval $(0,2\pi)$ [closed]

$\cos^2(\theta)-\sin^2(\theta)=1+\sin(\theta)$ over the interval $0<\theta<2\pi$ Find the trigonometric identity. Apologize for the confusion, first time using this resource didnt read the ...
1
vote
1answer
30 views

trigonometric identity domain restrictions for tan * cos

I'm having some difficulty in evaluating the domain restriction on this true/false problem: $$\tan(a) \cos(a) = \sin(a)\text{ for any }a \neq (2k + 1)\frac{\pi}{2}$$ I understand that the domain ...
3
votes
2answers
203 views

Continuity of sin x over rationals

I need a little help on the following question: For the function $f\colon[0, 2\pi]\to\mathbb R$ defined below, explain with proof, at which points of $c \in [0,2 \pi]$ $f$ is continuous or ...
0
votes
1answer
16 views

Rcosx alpha question

f(x)=3−4cosx+ 3sinx The question was to work out the maximum and smallest positive value is. I got this question wrong and I don't understand how to work out the maximum value.
0
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1answer
24 views

How do I show that the principal value of $\int_{- \infty}^{\infty}\sin(ax)\sin(bx)/x \,dx$ = 0

How do I show that the principal value of $\int_{-\infty}^{\infty}\sin(ax)\sin(bx)/x \,dx$ is equal to zero?
0
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0answers
24 views

Coordinates of a point between two other points with a exact distance from one of them

In this situation, I want the coordinates of the point C, I already know the coordinates of A and B, and the distance D (the distance between A and C). C must be on the A-B segment.
0
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2answers
48 views

What is sum of areas of all right triangles in first quadrant of unit circle?

I want to sum the areas of each right triangle formed as a radius sweeps out angles from $0$ to $\pi/2$ radians in the unit circle. Each triangle's area should be ...
0
votes
0answers
21 views

Weierstrass Trig Substitution Proof

After browsing some topics here, through one post, I discovered the "miraculous" Weierstrass substitutions. However, I can not find a decent or "simple" proof to follow. The simplest proof I found is ...
0
votes
0answers
32 views

Asymptotic analysis of Integrals of powers of sine and their application to intersections of hyperspheres

I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the ...
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0answers
9 views

Explain how to calculate the third vertex of a 2D equilateral triangle given two other vertices, using trigonometry

Before you comment that this has been asked multiple times, please read further. Given 3 arbitrary points: A(a, b), B(c, d), C(e, f), where AB = AC = BC, find C, knowing the values of A and B, using ...
0
votes
2answers
23 views

Fraction converted to PI

I've found this example: $$ 7\sin\left(\frac{2\pi}{5}(x + 1.25)\right) - 3 = 7\sin\left(\frac{2\pi}{5}x + \frac{\pi}{2}\right) - 3 $$ how could this $$(x + 1.25) $$ be converted to $$ x + ...
1
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4answers
24 views

How to substitute graph cosine and sine period?

for example the period of a normal function is: $$\text{period} = 2\pi $$ but in our graph the period is $$ \text{period} = 8\pi $$ to substitute it we make this: $$ f(x) =\cos\left(\frac14x\right) ...
1
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1answer
47 views

How to solve the exact value of this trigonometry

I want to know the exact value of cos50 Actually I have already tried lot of times to solve but I can not , find the exact value of cos50
0
votes
1answer
17 views

Finding out angle for Triangle with a Parallelogram inside of it?

I have a triangle with $\theta$ between $30^\circ$ and $60^\circ$. The opposite side has to be at least $4.2$ m and middle has to be $2.6$ m. How do I define the opposite side?
1
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2answers
50 views

How to sum up these sine functions? [duplicate]

What is the solution of this kind of exercise $A=\sin2a+\sin4a+\ldots+\sin10a$ I have already tried to multiply with $2\cos \frac a2$ to solve this.
0
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1answer
24 views

Trigonometry with quadrilateral and triangles

Find the value of $x$ Here's my solution: Note it is incomplete $$\begin{align} X\cos13&=AE\cos15\\ X\cos13&=(AF+AE)\cos15\\ X\cos13&=\left(\frac{1.5}{\sin15} + ...
0
votes
1answer
24 views

Determine midline equation of the function

$$ f(x) = -6\sin(3\pi + 4) - 2 $$ Why is $-2$ the midline of the graph of the function? How to prove that?
0
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3answers
86 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
0
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1answer
25 views

3D Vector defined by 3 angles trigonometry components

What I'm looking for is the trigonomery equations to calculate the x, y and z components of a 3D vector. What I mean: The counterpart formulas for a 2D vector defined by 1 angle: $x = ...
0
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1answer
33 views

Improving my method for evaluating the limit of this function

I could do with some help on the following question. \begin{equation*} \lim_{x\to \pi}\frac{\sin^2(x)-\tan^2(x)}{(x-\pi)^4} \end{equation*} I've calculated the result using l'Hopital 4 times, but ...
1
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2answers
42 views

Maxima and Minima of Sin(x)/x

I am trying to calculate the maximum and minimum points (between $-3\pi$ and $3\pi$) of $$f(x)=\frac{\sin(x)}{x}$$ I have found the derivative of the function and let it equal to zero. ...
0
votes
3answers
46 views

How to prove this trigonometrical inequality?

If $A=\sin ^8 \theta +\cos^{14} \theta $ then how to prove that for all values of $ \theta , 0< A \le 1$?
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2answers
43 views

Points of intersection of $\sin x$ and $\cos x$

I'm trying to find the points of intersection of $\sin x$ and $\cos x$ between $0$ and $2\pi$. I've tried but I keep getting 4 solutions... Would someone please be able to take me through the process? ...
2
votes
0answers
25 views

Simplifying Trig Identities

I need to factor and simplify $\cos(x)^4\sec(x)^2\cos(x)^2\tan(x)^4$. So far I know that $\cos(x)^4$ will cancel out because $\tan(x)^4$ becomes $(\sin(x)^4/\cos(x)^4)$ which leaves me with ...
4
votes
1answer
65 views

What did i do wrong with this derivation?

$$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} $$ Therefore \begin{align} \frac{1}{\cos(x)} &= \frac{1}{1-(\frac{x^2}{2} - \frac{x^4}{4!} + \frac{x^6}{6!} - \cdots)} \\ &= ...
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votes
1answer
30 views

Find all solutions in radians: $\sin(7x) = 1/2$ [closed]

Can someone give me a hint/guide me in the right direction for this problem please: Find all solutions in radians: $\sin7x = 1/2$
2
votes
0answers
17 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
0
votes
1answer
26 views

Finding the length with respect to x

This is a "simple" geometrry/trigonometry problem that I need for a physics problem and it is driving me insane because I can't figure it out. I have some particle that is moving along the x-axis ...
0
votes
2answers
70 views

How to solve $3\sin^3x-5\sin x\cos x+2\cos^2 x=0$?

Solve $3\sin^3x-5\sin x\cos x+2\cos^2 x=0$. It should use simple identities, but no identity I used helped me. There has to be a trick but I don't seem to find it. I could really use any kind of help. ...
1
vote
3answers
53 views

Why are trig functions defined for the unit circle?

Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles? If we apply the trig functions ...