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

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82 views

Trigonometrial proof

I'm trying to prove that $\sin(3\alpha) = 3\sin(\alpha) - 4\sin^2(\alpha)$. Using the angle sum rule, I've reached: $\sin(3\alpha) = \sin(2\alpha)\cos(\alpha) + \cos(2\alpha)\sin(\alpha)$ Hence, ...
1
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2answers
304 views

derive formula for height of tower on a hill

I'm working through a book to learn trig on my own and I got stuck with the following. This is the image given and the text in the book reads: Suppose you are standing an unknown distance away ...
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2answers
595 views

Derivative of integral of $\sin (t^2)$

I'm stuck with the problem If $ F(x)=\int_0^{x^3} \sin t^2 dt$ find $F'(x)$ Now, if the upper interval were $x$, the answer would be $\sin t^2$ (right?). However, the upper interval is $x^3$. ...
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1answer
43 views

For what value of $a$ equation $\cos2x +7 = a(2-\sin x)$ can have a real solution

Problem: For what value of $a$ equation $cos2x +7 = a(2-\sin x)$ can have a real solution In answer value of $a$ should be in the interval like $a \in(2,4)$ etc. Solution: $\cos2x +7 = ...
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1answer
101 views

Can someone please simplify this, please.

After solving my previous question, click here for question page, I tried to go up a notch and complicate the question just a bit further, turns out $\int e^x\sin(x)\cos(x)dx$ is much more different ...
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1answer
45 views

How do I calculate the intersection between two cosine functions?

$f(x) = A_1 \cdot \cos\left(B_1 \cdot (x + C_1)\right) + D_1$ $g(x) = A_2 \cdot \cos\left(B_2 \cdot (x + C_2)\right) + D_2$ Is it possible at all to solve this analytically? I can start ...
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2answers
40 views

What is the derivate of $A \cdot\cos{ (B\cdot (x + C))} - D$?

$$f(x) = A \cdot \cos{(B\cdot(x+C))} - D$$ $$f'(x) = \text{ ?}$$ I would assume that the derivate is something like this: $$f'(x) = - A \cdot B \cdot \sin{( \dots )}$$ The thing that troubles me is ...
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2answers
275 views

How can I find coefficients a, b, c given two points?

Suppose I have two points $A(x_A,y_A)$ and $B(x_B,y_B)$. How can I find coefficients $a, b, c$ of the straight line general equation ? $a x + b y + c = 0$
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2answers
118 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
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4answers
91 views

How to find the value of $\arctan(\frac{1}{1-x}) + \arctan(1-x)$?

I'm reading a book on complex analysis. In one step while evaluating a path integral, the author makes the following substitution: $$\arctan \left(\dfrac{1}{1-α} \right) + \arctan(1-α) = ...
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4answers
77 views

Find the value of $C$

We have a triangle $ABC$. Whats the value of angle $C$? $$\sin^2(A)+\sin^2(B)-\sin^2(C)=1$$ I made a small java program and it gave me an answer. I want to know how to make it through other ways.
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1answer
54 views

Please check my proof and let me know if incorrect.

$$\frac{1-\cos^2 x}{\tan x}= \sin(x)\cos(x)$$ i did the following working on LHS: $$\frac{\sin^2 x}{\tan x}=\frac{\sin(x)\sin(x)}{\tan x}=\sin(x)\cos(x)$$ i need to confirm that $$\frac{\sin x}{\tan ...
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3answers
166 views

Is this integral true? Or is it too much for Wolfram Alpha?

I was playing around with the wolfram calculator, just adding different things and mesmorising at what they equaled, then I randomly put in a bunch of trig functions, and well, this is what I got: $$ ...
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3answers
61 views

trig identity proof help working LHS only

$$(cos\theta - sec \theta)^2=\tan^{2}\left(\theta\right)-\sin^{2}\left(\theta\right)$$ I worked on LHS but not sure if I should put all of LHS in terms of sine/cos.
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1answer
143 views

Trig Bearing Problem

Help! This question is driving me mental! I am stuck on part b... Anyone have any hints please?
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1answer
88 views

How do I prove that both solutions to this differential equation $y"+k^2y=0$ are equivalent?

Consider the following differential equation $y''+k^2y=0$, where $y''$ is the 2nd derivative of y with respect to x. The solution to this equation is $y = A\exp(ikx) + B\exp(-ikx)$. However, another ...
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1answer
92 views

What does straight line general equation coefficients a, b, c mean

This is the straight line general equation: $\color{red}a x + \color{red}b y + \color{red}c = 0$ What does the coefficients $\color{red} a, \color{red}b, \color{red}c$ mean and what them names?
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4answers
181 views

Show that $\cot(5\theta)=\frac{1-10\tan^2(\theta)+5\tan^4(\theta)}{1-10\tan^3(\theta)+5\tan(\theta)}, \forall\theta\in R $

Show that $$ \cot(5\theta)=\frac{1-10\tan^2(\theta)+5\tan^4(\theta)}{1-10\tan^3(\theta)+5\tan(\theta)}, \forall\theta\in R $$ using De Moivre's theorem.
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5answers
73 views

trigonometry identity

I have some problem with proving this identity: $$2\left(1+\cos\alpha \right)-\sin^2\alpha=4\cos^4\frac{\alpha}{2}$$ I tried to start from the right side rewritting it to ...
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2answers
2k views

Taylor series for $\cot x$

Hi guys could you show me how to do the expansion of the Taylor series of $\cot x $ at the point $x=0$. My idea was to use $\dfrac{\cos x}{\sin x} $ and I want to expand it to the second term because ...
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2answers
2k views

Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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1answer
23 views

Given $\int\int_D \arctan \frac{y}{x}dxdy $ 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 $\int\int_D \arctan \frac{y}{x}dxdy $ 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 $r ...
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3answers
89 views

trig proof help please, my work is attached

I worked on both sides and thought i would end up with the original equation? my work is attached.
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1answer
47 views

Calculating the angle of a vector

Does anyone know how to total this up because im not sure, in order to find the angle. I'll greatly appreciate it. I need to know very soon! Exam tomorrow!
3
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4answers
108 views

very simple limit question $\lim_{t\to0}\frac{t^2}{1-\cos^2t}$

I'm trying to solve some limit problems and i found this one $$ \lim_{t \to 0}\frac{t^2}{1-\cos^2t} $$ what i did was, i changed $1-\cos^2t$ by $\sin^2t$ and i solve it like $$( \frac{t}{\sin t} ...
2
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1answer
43 views

Finding the least rational $r>0$ such that $\prod_{n=0}^3(2\cos(2^n\pi r)-1)=1$

Earlier a friend showed me a tricky problem he needed help with. I was able to find a possible solution but I've been unable to check it. Find the least rational $r>0$ such that $x=\pi r$ ...
6
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2answers
102 views

Taking repeated sin then cos

When you keep taking alternating sin and cos of any number as follows: $$\sin(\cos(\sin(\cos(\sin(\cos...(N))))...)$$ it seems to converge at about 0.69. Is there any way to find the exact value ...
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1answer
38 views

What is the domain for this function involving arcsin?

I am not sure what the domain of $\displaystyle \arcsin \left(x - \frac{2}{3}\right)$ is. I know that the domain for $\arcsin$ is $[-1, 1]$, but how can I use this to find the domain of the function ...
0
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1answer
57 views

How Can i find a point on a graph when i only have 1 point, angel, and units(distance)

How Can i find a point on a graph when i only have 1 point, angel, and units(distance)? Please consider me a novice with your answer? have been about 15 years since i was at school. It took me like ...
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3answers
137 views

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( ...
1
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2answers
169 views

Infinite derivatives of a trigonometric function

I've recently noticed that if you took an infinite amount of derivatives of a function, by that I mean something like this, $$ \lim_{x\to \infty} f^{'''\dots}(x)$$ Then if $f(x)$ is any polynomial ...
3
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3answers
118 views

How do you calculate the derivative after a change of variables?

How would you calculate $df \over dθ$ if $f(x,y) = x^2+y^2$ where $x = \sin 2θ$ and $y = \cos 2θ$? I tried Wolfram and using the product rule but I can't seem to get anywhere.
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4answers
562 views

Proof of the angle sum identity for $\sin$

$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$ How can I prove this statement?
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1answer
44 views

Proof using induction

I have no clue how to even start this: Proof using induction for every $k=1,2\dots n$ $$\vert\sin\sum_{k=1}^nX_{k}\vert\leq\sum_{k=1}^n\sin X_{k}$$ edit: Sorry, I also know this: $$0\leq ...
3
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2answers
82 views

Why does this assumption change the formula this way

I am working through some notes and I cannot understand why the following assumption changes the formula as such. The formula is basically referring to a right angled triangle of base $ L $ and ...
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1answer
45 views

Finding the derivative of a trig function

i'm having trouble a bit of trouble with taking the derivatives and collating my results of trig functions in the form of $sin$ $3x$ for example. The specific problem i'm stuck on is; Find the ...
3
votes
0answers
122 views

$\sin n$ having a closed form expression with no complex terms?

[All angles are in degrees] I have heard that we cannot express $\sin 1$ in closed form with no complex terms. However, I know that we can derive $\sin 18$ by solving $\cos 3x = \sin 2x$. Thus we can ...
0
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1answer
86 views

Tangent and arctangent functions: one-to-one?

Is it correct to state that arctangent function is one-to-one, but the tangent function is not? Or can this only be stated with imposed restrictions?
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4answers
2k views

calculator issue: radians or degrees for inverse trig

It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
2
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4answers
150 views

Solving $2\cos(x) = \sin(x)$

How would you solve equations of the form $ a \sin (x+b) = \sin (x)$? Eg. $ 2 \cos(x) = \sin(x) $ I realy have no idea how I would solve this kind of equations.
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3answers
78 views

Trigonometric inequality bounded by lines

How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this ...
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2answers
273 views

Prove that: $\sin{\frac{\pi}{n}} \sin{\frac{2\pi}{n}} …\sin{\frac{(n-1)\pi}{n}} =\frac{n}{2^{n-1}}$

Using that: $$ x^{n - 1} + x^{n - 2} + \cdots + x + 1 = \left(x - w\right)\left(x - w^{2}\right)\ldots\left(x - w^{n - 1}\right) $$ Prove that: $$ \sin\left(\pi \over n\right)\sin\left(2\pi \over ...
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2answers
100 views

Prove $p_0-p_2+p_4-\cdots=2^{n/2}\cos{\dfrac{n\pi}{4}}$ and $p_1-p_3+p_5-\dots=2^{n/2}\sin{\dfrac{n\pi}{4}}$

Consider: $$(1+x)^n= p_0 + p_1 x + p_2 x^2+\cdots$$ From where $$p_0=1,\quad p_1=\dfrac{n}{1},\quad p_2=\dfrac{n(n-1)}{2!},\ldots$$ Are the coefficients of the Newton´s Binomial expansion, using $x=i$ ...
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5answers
450 views

Why is $\tan((1/2)\pi)$ undefined?

Why is the trigonometric function $\tan((1/2)\pi)$ undefined?
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1answer
121 views

how to find sin13° cos13° tan13° cot13° with trigonometric circle.

I have problem finding sin(13°) cos(13°) tan(13°) cot(13°)with trigonometric circle. I have to draw the circle with a triangle on it but I can't get the right thing.
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1answer
52 views

Compute angle and radius of a circular segment

I need some help computing the angle and radius of a (given) circular segment. All I have is a start point $P_0 = (0,0)$ where the circular segment begins, at the origin. The length of circle arc ...
5
votes
2answers
86 views

Weird inequality

Let $x,y,z$ be real numbers such that $\cos x+\cos y+\cos z=0$ and $\cos{3x}+\cos{3y}+\cos{3z}=0$ prove that $\cos{2x}\cdot \cos{2y}\cdot \cos{2z}\le 0$.
5
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2answers
155 views

How find this $\sum_{i=0}^{5}\frac{1}{2+\cos{\left(x+\frac{i\pi}{3}\right)}}\cdot \frac{1}{2+\cos{\left(x+\frac{(i+1)\pi}{3}\right)}}$

Find this follow function $f(x)$ range ,where $x\in R$, $$f(x)=\sum_{i=0}^{5}\dfrac{1}{2+\cos{\left(x+\dfrac{i\pi}{3}\right)}}\cdot \dfrac{1}{2+\cos{\left(x+\dfrac{(i+1)\pi}{3}\right)}}$$ or ...
13
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5answers
569 views

Prove or disprove the implication:

Prove or disprove the implication: $a^2\cdot \tan(B-C)+ b^2\cdot \tan(C-A)+ c^2\cdot \tan(A-B)=0 \implies$ $ ABC$ is an isosceles triangle. I tried to break down the left hand side in factors, but ...
0
votes
1answer
29 views

Some trig identities wanted.

Let $S(x)=\sin^2(x)$ and $C(x)=\cos^2(x)$. Let $P_i$ be some multivariable polynomial with positive coefficients. Im looking for trig identities of the form $S(x)S(y)=P_1(S,C)$. And also ...