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Questions tagged [trigonometry]

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

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0answers
4 views

Spring model function with different amplitudes

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring t seconds ...
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1answer
18 views

Exponential decrease of amplitude with time

I was wondering about a particular math problem. It says that a particular trigonometric function, $10 \cos(2\pi x)$ models a bus going over a speed bump. They say that the amplitude decreases over ...
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0answers
35 views

How to work out this finite product?

$$\prod_{k=1}^{4n-2}\left[\sin\left(a+\frac{k\pi}{4n-2}\right)+\cos\left(a+\frac{k\pi}{4n-2}\right)\right]^{(-1)^k}\tag1$$ Suppose $a\ge0 $ and $n\ge1$ How to verify that $$(1)=(-1)^n\left[\frac{1-\...
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votes
3answers
58 views

Why does $\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi$? [on hold]

Show that $$\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi.$$
2
votes
1answer
42 views

Simplifying $\sin\frac{11\pi}{12}\sin\frac{29\pi}{12}-\cos\frac{13\pi}{12}\cos\frac{41\pi}{12}$. Why do I get the wrong answer?

Can someone explain why I get wrong answer in simplifying this expression? $$\sin\frac{11\pi}{12}\sin\frac{29\pi}{12}-\cos\frac{13\pi}{12}\cos\frac{41\pi}{12}$$ If we rewrite the expression with new ...
3
votes
1answer
52 views

Calculating $\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + …+\frac{\sin(2\sqrt n)}{n\sqrt n\cos\sqrt n}\right)$

Using the trigonometric identity of $\sin 2\alpha = 2\sin \alpha \cos \alpha$, I rewrote the expression to: $$\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + ...+\...
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0answers
15 views

Find a point on a circle which contains a rectangle using another point, the angle between the two and the rectangle's dimensions

I'm trying to construct a mathematical formula that will calculate a point (x,y) on a circle which contains a rectangle with a width ...
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2answers
41 views

Trig Identity Question Finding Value of K

If $\sin(x) + \cos(x) = k$ for what value(s) of $k$ can $\sin(x)\cos(x)=1$?
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4answers
19 views

Parametrization of a line segment using angle as parameter

I know this is probably elementary level for most people here, but I've been stuck on this problem for no less than 4 hours and I am completely clueless as to how to figure this out. Is it possible ...
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1answer
39 views

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$ $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to?

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$, $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to ? My thinking:- I have break the left hand side on $sinC + sinD$ and right hand ...
35
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1answer
4k views

Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

I was browsing through a Wikipedia article about the trigonometric identities, when I came across something that caught my attention, namely forgotten trigonometric functions. The versine (arguably ...
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0answers
25 views

Circumference touching a sine wave

I'm trying to get the intersection points between a sine function and a circumference. So, i have this equations: $y = a\sin(bx + c) + d$ $(x-h)^2 + (y-k)^2 = r^2$ If i substitute the sine in the ...
2
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2answers
30 views

Two possible angles in a triangle

"Determinate a value for $\angle DBC$ if $\angle DAC = 2\angle DCA= 40º$ and $BC=\sqrt 3\space AD$. "The diagram is not to scale" After trying this problem, i ended with $\sin \angle DBC = \cos 20º$ ...
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1answer
32 views

Trigonometry: Model of snowfall

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. a) Construct a function that models this behavior. b) During what ...
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1answer
46 views

Is $\tan^{-1}(-1) = 3\pi /4$ or $=7\pi /4$? I understand they're both valid solutions, but what about places where the value is added/subtracted?

For example: calculate $\int^4_2 \tan^{-1} x \, dx$ If $\tan^{-1}(-1) = 3\pi /4$, then the final answer is $\frac{-\pi}{2}$. But if $\tan^{-1}(-1)= 7\pi /4$, then the final answer would be $\frac{3}{-...
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0answers
20 views

How to find the center of an after-rotated rectangle in 2d-space?

I'm trying to write an algorithm that solves the following problem. And although there is a lot of rectangle geometry questions here on math.stackexchange.com, I have not yet found one that answers ...
4
votes
2answers
78 views

Simplifying $\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$

I am looking to simplify the following, without the use of capital Pi notation: $$\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$$ Which is meant to produce the sequence: $\left[1,\ \frac{1}{2},\ \...
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0answers
26 views

Extract param of $\sin$ from expression $y=2(\sin b-\sin a)/(\sin c-\sin a)$

$y=2\frac{\sin b-\sin a}{\sin c-\sin a}$, where $a=q(n+0)$ $b=q(n+1)$ $c=q(n+2)$ $q=\frac{2 \pi f}{s}$ Is it possible to extract $n$ from this formula? I already try this on WolframAlpha but I do ...
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3answers
48 views

Solving Trigonometric Questions Without a Calculator [on hold]

How do I solve the following question without using a calculator?
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2answers
45 views

Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. [on hold]

Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. Which half-identity formular should I use and why?
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1answer
41 views

Find $\tan \left(\frac{\theta}{2}\right)$, given $\sin (\theta) = \frac 35$, with $90^\circ < \theta < 180^\circ$ [on hold]

Find tan theta/2, given sin theta = 3/5, with 90^∘ < theta < 180^∘. I don't know how to solve it! Help? do I use the tangent half-identity formula?
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2answers
44 views

Find the median in a triangle with trigonometry

In a triangle $ABC$, $AB=7$, $AC=4$ and $\angle CAB=50º$. Let $M$ be the midpoint of $BC$. Determinate $AM$. My try I applied law of cosines $3$ times, first to find $BC$, then I let $\angle BCA=\...
2
votes
1answer
98 views

Inverse of $\frac{\sin(x)}{x}$

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a ...
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votes
0answers
19 views

find the curve of best fit of the type $y= a \sin(bx)$ by the method of least squares [on hold]

Find the curve of best fit of the type $y= a \sin(bx)$ by the method of least squares?
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0answers
14 views

Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]

Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following: \begin{equation} \sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0 \end{equation} I tried to use Euler ...
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2answers
24 views

A puzzle about replacing $v_1$, $v_2$, $v_3$ while retaining the linear independence of the resulting set.

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section3.5 Exercise 5, I was puzzled at some of it. Here is the problem description: Exercise 5. ...
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0answers
16 views

What would the law of the tangents be for a tetrahedron?

In generalised trigonometry, a corollary for the law of sines in a tetrahedron with vertices A,B,C and D is defined as being: sin (angle DAB) multiplied by sin (angle DBC) multiplied by sin (angle DCA)...
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0answers
23 views

How to find the location of a point in 3D space from projected 2D angle

I have points $A,B,C$ in 3D space and I know the position of $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. I want to find the location for $C$ given that $AB$ and $BC$ is perpendicular in 3D space but ...
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votes
1answer
35 views

Find the general value of $\theta$

Find the general value of $\theta$ which satisfies the equation $(\cos\theta+i\sin\theta)(\cos3\theta+i\sin3\theta)\dots \{\cos(2n-1)\theta+i\sin(2n-1)\theta\}=1$. My attempt: $(\cos\theta+i\sin\...
1
vote
3answers
41 views

Prove $ \frac{\sin\theta}{1-\cos\theta} - \frac{\sin\theta}{1+\cos\theta} = 2\cot \theta$

Prove $$ \frac{\sin\theta}{1-\cos\theta} - \frac{\sin\theta}{1+\cos\theta} = 2\cot \theta$$ So I started by combining the two fractions, which gave me: $$ \frac{\sin\theta(1+\cos\theta) - \sin\theta(...
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2answers
25 views

Exponential double angle formula

My question is whether someone could provide a proof for the following identity: $$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$ Motivation: The left hand side is ...
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4answers
53 views

Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$

Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ My apporach:- $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ $$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
4
votes
1answer
65 views

Can one simplify $\arctan(a\tan(x))$?

We know that $\arctan(\tan(x))=x$ when $x$ lies between $-\pi/2$ and $+\pi/2$; but do you know a way to transform the expression $\arctan(a\tan(x))$, where $a$ is a real number between $0$ and $1$? I ...
2
votes
4answers
66 views

If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.

If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$. My approach:- $$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \...
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votes
1answer
40 views

how do you find the maximum and minimum value of cos x - 3 sin x = y [on hold]

how do you find the maximum and minimum value of $\cos\theta - 3\sin\theta = y$
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0answers
42 views

Summation Formula for Tangent/Secant Numbers

I came across the following expressions: $$\begin{align} \widehat{S}_{2n} &:= \sum_{1 \leq k_1<\cdots<k_n \leq 2n} \prod_{\ell=1}^n (k_\ell-2\ell)^2, \\ \widehat{T}_{2n+1}&:=\sum_{1 \...
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5answers
54 views

If $\frac{a}{\sin{A}}=\frac{b}{\cos{A}}$, show that $\sin{A}\cos{A}=\frac{ab}{a^2+b^2}$

I don't know how to go about solving this, I think I need to use $\sin^2\theta+\cos^2\theta=1$, but I'm not sure how to go about this. The closest I managed to get was: $$\frac{a}{\sin{A}}=\frac{b}{\...
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votes
1answer
37 views

Solving $2\tan(x-15) = 3.7.$

I don't understand the reason for the method to answer the question below. Why is it possible to subtract 180 from 61.6, rather than adding by 180? $$2\tan(x-15) = 3.7,\quad -180\leq x\leq 180.$$ 1)...
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1answer
46 views

Sum of finite cosine functions

Is it possible to apply trigonometric conversion on multiple sum of cosine functions ? I have a sum like this: $$\cos(\pi a_1) + \cos(\pi a_2) + \cos(\pi a_3) + ... + \cos(\pi a_{10})$$ The ...
2
votes
2answers
29 views

Finding the angle between a line and a plane

Given that the equation of the line is: $$ \mbox{P:}\quad \left\{\begin{array}{rcrcrcr} 3x & - & y & + & z & = & 6 \\ x & + & 2y & + & z & = &-3 \end{...
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2answers
24 views

Number of solution for $\sin x = 0$ in $[0,2π]$ are 2 or 3

Number of solution for a equation $\sin x = 0$ in $[0,2π] $ (close interval) are 2 or 3. Solutions are of course 0,π,2π but are they three solution or just two solution considering 0 and 2π are same ...
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1answer
85 views

A question about a fly and a spider

A spider is sitting exactly in the middle of one of the smallest walls in a living room, whilst a fly is resting by the side of the window of the opposite wall, 1.5 m above the ground and o.5 m from ...
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votes
1answer
35 views

How to find equation to find a,b,c from below linear equations Given inputs x,y,z and angles \theta,\alpha? [closed]

$$x=a\cos\theta-c\sin\theta$$ $$y=(c\cos\theta+a\sin\theta)\sin\alpha+b\cos\alpha$$ $$z=(c\cos\theta+a\sin\theta)\cos\alpha-b\sin\alpha$$ Need equation for a,b,c
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3answers
71 views

How to solve $2 = x \tan(x)$?

When I plot the equation $2=x\tan x$ on Mathematica, I am seeing multiple zeros. But I am stumped as to how to find these zeros on paper.
3
votes
1answer
46 views

Why is this limit evaluated like so?

Question: If $$\lim_{x \to 0}{\frac{-1 + \sqrt{(\tan x - \sin x) + \sqrt{(\tan x - \sin x) + \sqrt{(\tan x - \sin x) + \cdots \infty}}}}{-1 + \sqrt{x^3 + \sqrt{x^3 + \sqrt{x^3 + \cdots \infty}}}}} = \...
0
votes
0answers
13 views

Asymptotic number of peaks in a large product of sums of oscillatory functions

I am interested in the following asymptotic question. Suppose I have the function $$g_N (\alpha_0) = \frac{ \prod_{i=0}^{N-1} \left[\;c_+ (N-i) + c_- (N-i)\cos(2\alpha_0)\;\right]}{\int_0^{2\pi} d\...
0
votes
2answers
38 views

What would $C$ be if I'm trying to solve it in a tangent inverse $\tan^{-1}(C) = \frac{\pi}{4}$?

I am trying to figure out what would be $C$ when the problem reads $$\tan^{-1}(C) = \frac{\pi}{4}?$$ I've struggle with problems like these all the time. Calculator won't help me with these.
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votes
1answer
41 views

Is tan(y) = x + C, the same as Y = x/tan + C/tan? [closed]

Alright so the title says it all. I am working on a method called Separation of Variables with differential equations. The original problem was Y'= cos^2y. I am wondering since I was trying to get ...
1
vote
2answers
54 views

Finding the interior angles of an irregular polygon inscribed on a circle

Is there any way to calculate the interior angles of an irregular N-sided polygon inscribed on a circle? I only have a list of edge lengths (in order). I don't know any of the interior angles nor the ...
1
vote
2answers
101 views

What is $\sin(2)$ equal to?

I understand that $\sin(2)$ is equal to $0.90929\ldots$. However, I am wondering if there is a simpler way of calculating this. Essentially I am wondering if there is an answer with this where $\pi$ ...