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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is tan(π/2 - x)=cot(x) because tan in the 2nd area is "negative" and also cot(-x) equals to -cot(x)?
trying to figure out if that negative/positive calculation is true.
and so at the end there will be two negative and the cot(x) will be positive.
is that so?
thank you.
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Proof without words for the sum and product of three tangents
How to get the $\tan\beta \tan\gamma$ in the third graph?
Proof without words for the sum and product of three tangents
The image is original from MAA
https://www.maa.org/sites/default/files/...
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Find a triangle's vertices given sides, angles, and inscribing circle
I have a triangle inscribed in a circle of radius 1 centered on the origin. The angles A B C and opposite sides a b c of the triangle are given (but variable). Given the coordinates of one vertex (0,-...
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Prove 2D rotation by $\theta$ is a linear transformation using trig identities for $r\cos(\alpha) + s\cos(\beta)$ and $r\sin(\alpha) + s\sin(\beta)$
Theorem
The rotation by $\theta$ function $T:\mathbb{R}^2 \to \mathbb{R}^2$ defined for $\vec{x}%=\begin{bmatrix}x_1 \\ x_2\end{bmatrix}
=\begin{bmatrix}r\cos(\alpha) \\ r\sin(\alpha)\end{bmatrix}$...
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exponential equation for the distance between two points a distance away from the function [0,1,...,9]x
Given the element list: b = [0,1,...,9]
Then what is the exponential equation in relation to distance from two points with a value of d from the origin of the functions: $(b_n)x$ and $(b_n-1)x$.
To ...
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System of equations with sin and cos
So im currently trying to find the extremas of the function $f(x,y) = \cos(x+y) + \sin(x) + \sin(y)$
I've already computed the partial derivatives:
$$
f_x(x,y) = \cos(x) - \sin(x+y)\\
f_y(x,y) = \cos(...
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Which angle does the calculator give for $\arctan$? [closed]
I know that there are two possible angles for the $\arctan$ (that is, $\tan^{-1}$) function.
When I do $\arctan$ using a calculator, do I get the smaller angle? Or are there other criteria for this?
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Pls help with this gr 10 math problem :( [closed]
A 200 m tall communication tower is secured to the ground using 8 guy wires. Four long wires are attached to the top of the tower and make an angle of 63" with the ground. Four shorter wires are ...
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Solution of system of nonlinear equations with trigonometric terms
Issue:
I am trying to solve the following system of nonlinear equations for the unknown variables: $x$, $z$ and $\beta$. The remaining variables are known values.
$$a=u(s^2+(x\cos\beta\ )^2+(z\sin\...
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Rotating $y=x-2\sqrt{x}+1$ by $45^\circ$ counter-clockwise
I tried to rotate the equations
$$y=x-2\sqrt{x}+1 \quad\text{and}\quad y=x+2\sqrt{x}+1$$ (because of the two possible answers) $45^\circ$ counter-clockwise by using
$$y \cos(\theta)- x \sin(\theta)=(f(...
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Series expansion with nth term of $\left(\frac{\sin (x)}{x}\right)^a$
Using Mathematica, I need an expansion with $n$th term of $$ f(x)=\left(\frac{\sin (x)}{x}\right)^a $$ about $x=0$ where $a\geq 0$ or if $$f(x)=\sum_{n=0}^{\infty} b_{2n }x^{2n}$$ then I need a ...
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Can I infer the hypotenuse given only $a > b$?
Today, I came up with a problem. The problem is this:
Let $a$, $b$ and $c$ be the sides of a right-angled triangle, such that $a > b$. The nature of $c$ is unknown. Can I infer the hypotenuse given ...
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Are there inverse trig compound angle identities?
There are well known identities for compound angle trigonometry, but I can't find anything for inverse trig compound angles, and haven't been able to derive any myself. Are there identities for:
...
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How to solve $\,A\sin(\theta_2-\theta_1) - B\sin(\theta_1) = 0$
I want to find the solutions of the following equation (In order to find the singular points of a robot).
$A,B$ are positive numbers, and actually :
$A = 0.2531,$
$B = 0.2455.$
$$ A\sin(\theta_2 - \...
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Help with determining constant values for water depth measurements using trigonometric functions
I am currently working on a problem related to water depth measurements and I am seeking some help with it. The problem is as follows:
Water depth at a dock is measured during the first 16 hours of a ...
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Find the sides of an AAA triangle inscribed in a circle with known radius
I have all of the angles for a triangle. I want to know the length of each of the sides such that the three vertices will all lie on a circle with a known diameter. How do I do it?
(The application is ...
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Proving $\cos(\pi/2^n) =\frac{1}{2}\sqrt{2+\sqrt{2+\cdots\sqrt{2}}}$ [duplicate]
WolphramAlpha gives the following identities for the cosines of $\frac{\pi}{2^n}$:
$$
\cos\left(\frac{\pi}{8}\right)=\frac{1}{2}\sqrt{2+\sqrt{2}}
$$
$$
\cos\left(\frac{\pi}{16}\right)=\frac{1}{2}\sqrt{...
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Find minimum and maximum of $A\sin x+B\cos x+C\sin x\cos x$
Find minimum and maximum of $(A\sin x+B\cos x+C\sin x\cos x)$ where $A,B$ and $C$ are real numbers
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In the Law of Cosines, $a^2=b^2+c^2-2bc\cos\theta$, what does the $-2bc\cos\theta$ term do?
The Law of Cosines
$$a^2=b^2+c^2-2bc\cos\theta$$ looks like an extension of the Pythagorean theorem, but what does the second part ($-2bc\cos\theta$) actually do?
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how do you prove that sin is a function of arclength(radian) to y coordinate?
I was watching https://youtu.be/TpWQlKHPyJ4
what it does is the following
calculate the arclength of a unit circle in terms of x
integrate $(1-x^2)^{-1/2}$
you get arclength = $sin^-1$(x)
and the ...
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Find the measure of the smallest positive angle $\theta$ in degrees for which $\tan\theta=\frac{\cos25^\circ+\cos85^\circ}{\sin25^\circ-\sin85^\circ}$
I'm preparing for a math competition, and was stumped by this problem. The original problem is shown below, and the correct answer is $120°$. I'm posting this here to ask for explanation on how this ...
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Is there a way to determine a proof for the expansion of $\sin(z)=x$ where $\vert x\vert\gt-1$ [closed]
Is there a way to determine a proof for the expansion of $\sin(z)=x$ where $\vert x\vert\gt-1$
So I want to find a way to write a proof that proves the expansion of $\sin(z)=x$ for $x$ where $x\in \...
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Lagrangian of a spherical pendulum
I am working on a problem that's asking me to express the lagrangian of a mass $m$ suspended from a rigid massless rod of length $L$, but free to rotate otherwise (a spherical pendulum). The problem ...
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Why do sometimes , the answers I derive from equations not actually provide solution in inverse trignometry?
This is the question:
Solve for $x$
$$\arcsin(1-x)-2\arcsin(x) = \frac{\pi}{2}$$
I solved this by these steps:
$\arcsin(1-x) = \frac{\pi}{2} + 2\arcsin(x)$
$\sin$ function on both sides
$1-x=\sin(\...
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Product of trigonometric functions as a trigonometric polynomials
We are given a non-negative trigonometric function $f = (\cos^2(\theta))^n(\sin^2(\theta))^m$,
where $m=N-n$ and $n,m<N$. I would like to understand if such a function can be reshaped to look like ...
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How to evaluate the following summation?
I have managed to solve the first half of the question by using A=kπ/2n and B=(k-1)π/2n, furthermore,How do you solve the second part of the question?
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Evaluating $\cos\frac{\pi}{7}\,\cos\frac{2\pi}{7}\,\cos\frac{4\pi}{7}$ [duplicate]
$$\cos\frac{\pi}{7}\;\cos\frac{2\pi}{7}\;\cos\frac{4\pi}{7}$$
I reckon we can use some trigonometric equations but I can't figure out how, I tried sum to product equations but I can't find the answer ...
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If $x,y∈(-π,π]$, then find the area of the polygon formed by points $(x,y)$ satisfying the equation $\lfloor|\sin x|\rfloor+\lfloor|\cos y|\rfloor=2$.
If $x,y∈(-π,π]$, then find the area of the polygon formed by points $(x,y)$ satisfying the equation $\lfloor|\sin x|\rfloor+\lfloor|\cos y|\rfloor=2$.
My attempts include using a graphing tool and ...
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Find the condition such that $A \cos{x} = x$ has exactly two solutions.
This is something we all do in high school but I forgot how to solve such a problem. It recently came up in my theoretical Physics research. I want to find a constraint on the variable $A$ such that ...
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Inequality of $|\sin(a+b)-\sin(c+d)|<|(a+b)-(c+d)|$
I am reading a solution with this inequality:
$$|\sin(a+b)-\sin(c+d)|\le|(a+b)-(c+d)|$$
The solution just says this holds, but I don't quite understand how?
I am also trying to know if it can be ...
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Expand $\cos^a x$ in terms of $\cos kx$, $\sin mx$
If $a\geq 0$, expand $\cos^a x$ in terms of $\cos kx$, $\sin mx$
$$\cos^a x=\left(\frac{e^{ix}+e^{-ix}}2\right)^a$$
Since $a$ is a non negative real number, so by General Binomial theorem
$$\cos^a x=...
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How to find minimum degrees required to get X distance from the far left side of a circle?
I'm trying to figure out how far each degree of a circle is, assuming the far left side of a circle is x=0. To put it another way I need to figure out how far each degree of the circle takes it from ...
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How to prove $(\sec^2(a)+\tan^2(a))(\csc^2(a)+\cot^2(a))=1+2\sec^2(a)\cdot\csc^2(a)$ [closed]
How can I prove that
$$\left(\sec^2(a)+\tan^2(a)\right)\left(\csc^2(a)+\cot^2(a)\right)=1+2\sec^2(a)\cdot\csc^2(a)?$$
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Describe in a trigonometric manner the radius of circle ABD.
Let $D\in BC$ in $\triangle ABC$, so that $AD$ is the bisector of $\angle BAC$. Let $r$ be the radius of the circle inscribed in $\triangle ABC$ and $R$ the radius of the circumscribed circle of $\...
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Finding angle and radius of arc with one known and one partially known point
Given two points $p1$ and $p2$ on a circle of radius $R$ and knowing the following:
Arc length $L$
$p1$ (which is simply the origin at 0,0)
The $y$ coordinate of $p2$ (but not the $x$ coordinate)
Is ...
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Prove that the sum of division of two squared sin function is 1
When I numerically compute the below summation it is always $1$. How can I prove this? $N$ is an integer number and $k$ is an integer number between $0$ to $N-1$ and $x$ is real number between $0$ and ...
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local Coordinate transformation to global coordinates
Reference image
Please help me to prove;
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What is the perímeter of the triangle ABC?
I am froze in this question after some hours.
It's a homework (without a detailed solution in the book) that I am trying to do.
I am graduated in high school, but there is some years I didn't study ...
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How does $\cos^{-1} (1/\sqrt2) = \pi/4$ ? I thought $\cos(\pi/4) = \sqrt2/2$
How does $\cos^{-1} (1/\sqrt2) = \pi/4$ ? I thought $\cos(\pi/4) = \sqrt2/2$
I am awful at formatting, thank you ahead of time.
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Can you tell me the reason behind this equation , written below [closed]
-cot(x+pi/3)=tan(pi over 2 plus x plus pi over 3)
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Trigonometric identity $(\csc \theta - \cot \theta )^2 = \frac{{1 - \cos \theta }}{{1 + \cos \theta }}$ [closed]
I was doing some problems related to trigonometry but I couldn't figure out how to do this sum. I really wanted the help to figure it out. Thanks very much in advance.
$$
(\csc \theta - \cot \theta )^...
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We have to find maximum value of the expression. [closed]
enter image description here
I have tried to solve this problem by taking derivative on both the sides but solving algebra and trigonometry simultaneously doesn’t provide a neat solution. Plotting the ...
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Understanding the trigonometry behind a pole leaning on a wall
I am copying and pasting this question I asked in the physics stack exchange. The reason I am doing so is because the question was automatically closed for being too "homework-like" and not ...
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Find the solutions of $x$ such that $\arcsin x=2\arctan x$
Find the solutions of $x$ such that $\arcsin x=2\arctan x.$
My solution goes like this:
We have, $\arcsin x=2\arctan x.$ Now, $2\arctan x=\arctan(\frac{2x}{{1-x^2}})=\arcsin(\frac{2x}{1+x^2}).$ Now, ...
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Proving $\sum_{k=1}^{2n-1}\frac{\sin(\frac{\pi k^2}{2n})}{\sin(\frac{\pi k}{2n})}=n$
I wander on the internet and found this problem (from Quora) this link
The problem is proving the identity: $$\sum_{k=1}^{2n-1}\frac{\sin\left(\frac{\pi k^2}{2n}\right)}{\sin\left(\frac{\pi k}{2n}\...
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How can I briefly but concisely explain the unit circle to my daughter? [closed]
My daughter is currently in Geometry, but she has already worked some with beginner concepts of Trigonometry. She understands SOHCAHTOA. The unit circle came up in conversation, and I attempted to ...
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Solid angle on a sphere of the intersection of multiple hemispheres
Consider the following scenario where I have a unit sphere cut by $N$ halfplanes that all contain the center of the sphere, thus forming several hemispheres. How can I calculate the solid angle ...
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trigonometric identity series [closed]
enter image description here
I am looking at complex analysis and I need to prove this series using complex algebra recording the identity 1+z^1+z^2+z^3+...+z^n=(1-z^(n+1))/( 1-z)
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Inscribed triangle in a circle from perpendicular chords
I'm trying to solve the following problem but have hit a wall.
Let's assume we have a circle. Said circle has a chord from point $A$ to point $B$. At one of the points where the chord crosses the ...
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Choosing optimal sample points for cubic spline approximation of sinusoid with polynomial argument
My problem consists as follows. Say I have a sinusoidal function whose argument is a cubic polynomial:
$$f(t) = \sin(at^3+bt^2+ct+d) \quad a, b, c, d\in \mathbb{R}, \quad t\in[T_1, T_2]$$
I want to ...
