Questions tagged [trigonometry]
Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.
30,561 questions
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Isoperimetric quotient of sec x+ sec y= sec fraction pi the square root of 2 +1 via single Fourier?
Correct?
$R=\pi/\sqrt{2},\quad c=\sec R+1,\quad n=8.$
Axis. $\sec R+\sec 0=c$ and $R^2+0=R^2$, so $r(0)=R$.
Pole. $x=\tfrac\pi2+u,\; y=\tfrac\pi2-v,\; u,v\to0^+$:
$$\sec\!\left(\tfrac\pi2+u\right)\...
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Why do two surface points on a reference ellipsoid appear to admit two valid connecting curves instead of one?
Suppose two surface points (A) and (B) lie on an oblate reference ellipsoid.
If I represent the points using geocentric geometry (position vectors from the ellipsoid center), the construction appears ...
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Angle relations in triangle
Why does this
$$h_z = \begin{cases}
-z(L+x) \; , & x\in \left[-L,-z\right]\\
(L-z)x \; , & x\in \left[-z,z\right]\\
z(L-x) \; , & x\in \left[z,L\right]
\end{cases}$$
define a $2L$-periodic ...
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What is the TOUGHEST problem ever found in Trigonometry? [closed]
What do you think that it could be the TOUGHEST problem ever found in Trigonometry?
I'm very curious to know about.
A note to the readers:
The absolute TOUGHEST problem can't be found, but there can ...
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Prove the reciprocal of $AB$ is equal to the reciprocals of $AC$ plus $A'C'$
Suppose $A,B,C,A',B',C'$ are six points in the plane such that $AB=A'B'$, and $\angle BAC=\angle B'A'C'=60^{\circ}$ and $\angle ABC + \angle A'B'C'=180$.
Prove that $ \frac{1}{AB}=\frac{1}{AC}+\frac{1}...
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Euclidean distance between two points on a cone
So I have a cone $C$, with two points $a$ and $b$ on which we have access to the following information:
the radius $r$ of the base of the cone
the respective distances $d_1$ and $d_2$ from each point ...
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Is there a more efficient method to calculate arctangent to arbitrary precision?
I am an unemployed individual who is a high-school dropout very good at programming, I write programs just for fun and I have implemented many ways to calculate arctangent to arbitrary precision, all ...
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What are the two possible functions from red shift to time?
On an expanding $S^n$ sphere, one can perceive doppler effect of a wave medium travelling towards oneself and thereby measure the time $t$ since the wave was emitted as a function of the frequency ...
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Determinant of a $2026 \times 2026$ matrix of sines
From the first round of the 2026 Mexican University Math Olympiad
Determine the determinant of the $2026 \times 2026$ matrix $A$ whose entries are defined by $$ A_{i,j}=\sin(2026i+j),$$ for $1\le i\...
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Why is this curve almost a circle? $~\sec x+\sec y=\sec\frac{\pi}{\sqrt2}+1$
The blue curve below is the circle $x^2+y^2=\dfrac{\pi^2}{2}$. The red curve is $\sec x+\sec y=\sec\dfrac{\pi}{\sqrt2}+1$.
$~~~~~$
Below they are shown on the same set of axes. (It looks like a single ...
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Find the area of a triangle given one side and angle and the sum of the other two sides
I'm stuck in the following problem. I don't remember where I found it, but is kind of an Olympiad problem.
Let $ABC$ be a triangle where $a=\sqrt{6}, \hat{A}=\pi/6$ and $b+c=3+\sqrt{3}$. Fin the area ...
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How can we build connections between trigonometric functions and hyperbolic functions through complex exponential functions? [duplicate]
As we know, the trigonometric functions $\cos x$ and $\sin x$ appear as the real and imaginary parts of $e^{ix}$, while the hyperbolic functions $\cosh x$ and $\sinh x$ are naturally related to $e^x$.
...
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Show that $\int_0^{\pi/2}\left(\frac{x}{(\cos x+1)\sin x}+\frac{(2-\cos x)\sin x-x}{\sin^3 x}\right)dx=1+\frac{\pi}{4}$
Let
$$I=:I_1+I_2$$
where
$$I_1=:\int_0^{\pi/2}\frac{x}{(\cos x+1)\sin x}dx$$
$$I_2=:\int_0^{\pi/2}\frac{(2-\cos x)\sin x-x}{\sin^3 x}dx$$
Show that $$I=1+\frac{\pi}{4}$$
Help from Wolfram
Wolfram ...
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Why does $m = 2k+1$ maximize $\sum_{p=0}^{m-1} \sin\frac{2\pi p}{k}\sin\frac{4\pi p}{m}$?
Let $k$ be an odd positive integer, $k \geq 3$. For odd integers $m > 2k$, define:
$$S(m) = \sum_{p=0}^{m-1} \sin\!\left(\frac{2\pi p}{k}\right) \sin\!\left(\frac{4\pi p}{m}\right)$$
Numerical ...
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Determining the range of $f(x) = \sin^n x + \cos^n x$ for integer $n$ [closed]
Is it possible to find some kind of formula to determine the range of the following function $$f(x) = \sin^n x + \cos^n x$$ for all positive integers $n$?
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