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 Yearling
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1h
comment Calculation of integral using two different methods?
To complete the discussion and show that the two solutions are consistent with each other, subtract one from the other and note that the difference is a constant.
4h
answered Trying to solve a pair of trigonometric simultaneous equations
7h
comment Sum/Difference Identity Formula Question
You have the right idea. However, the equation has $13\pi/15$, whereas your solution uses $13\pi/5$ (that is, the denominators are different). Based on the equation as written, the solution should be $$\cos\left(\frac{13\pi}{15}+\left(-\frac{\pi}{5}\right)\right) = \cos\left(\frac{13\pi}{15}-\frac{3\pi}{15}\right) = \cos\frac{10\pi}{15}= \cos\frac{2\pi}{3}$$
1d
comment Prove that a trigonometric equation has six distinct roots
Descartes' Rule of Signs also indicates problematic cases. For instance, if $C+B$, $C-B$, and $A$ are all positive, then there are either $2$ negative roots, or none at all. Or, if $A=0$, while $C+B$ and $C-B$ match in sign (but aren't themselves zero), then there are no real roots. And so forth. The possibility of six roots requires that $B+C$ and $B-C$ have different signs.
1d
comment Given 4 points with 2 on different radius. Obtain the center of the circle.
What geometric relationship exists between the black hole and two positions of a particular star? (In other words: How does the center of a circle relate to two points on that circle?)
1d
revised Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$
deleted 122 characters in body
1d
revised Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$
added 629 characters in body
1d
answered For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?
1d
comment Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$
@sepideh: Actually, there's another solution.
1d
answered Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$
1d
comment Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$
Have you tried anything?
2d
revised Trig identity involving sum of cosines
added 46 characters in body; edited title
2d
comment What is the equation describing a three dimensional, 14 point Star?
@ThePolywellGuy: "The actual solution is:" ... well?
2d
awarded  Yearling
2d
comment Need help with a design calculations equal spacing of circles
I'm not sure I understand your question. Could you provide a picture of a sample situation?
Jul
27
revised For acute $\theta$, write $\cot\theta$ in terms of $\sin\theta$
Better title, TeX improvement
Jul
26
comment Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
I feel compelled to mention the Laws of Cosines for tetrahedra. Writing "$X_2$" for $X/2$, and "$\angle XY$" for the dihedral angle between faces $X$ and $Y$, Euclid has $$W^2=X^2+Y^2+Z^2-2XY\cos\angle XY-2YZ\cos\angle YZ-2ZX\cos\angle ZX$$ Hyperbolic/spherical space has $$\begin{align}\cos W_2&=\cos X_2\cos Y_2\cos Z_2\pm\sin X_2\sin Y_2\sin Z_2\sqrt{S}\\&+\cos X_2\sin Y_2\sin Z_2\cos\angle YZ \\&+\sin X_2\cos Y_2\sin Z_2\cos\angle ZX\\&+\sin X_2\sin Y_2\cos Z_2\cos\angle XY\end{align}$$ where $$ S:=1-2\cos\angle XY\cos\angle YZ\cos\angle ZX-\cos^2\angle XY-\cos^2\angle YZ-\cos^2 \angle ZX$$
Jul
26
comment Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
The disparity grows in higher dimensions. If $W$ is the "hypotenuse-face" of a right-corner tetrahedron, and $X$, $Y$, $Z$ the "leg-faces", then Euclidean space has $$|W|^2 = |X|^2+|Y|^2+|Z|^2$$ (where "$|\cdot|$" is face area), but non-Euclidean space has $$\cos\frac{|W|}{2}=\cos\frac{|X|}{2}\cos\frac{|Y|}{2}\cos\frac{|Z|}{2}\pm \sin\frac{|X|}{2}\sin\frac{|Y|}{2}\sin\frac{|Z|}{2}$$("$-$" for hyperbolic, "$+$" for spherical). Indeed, in dimensions $4+$, Euclid always has $$|\text{hyp}|^2=|\text{leg}|^2+|\text{leg}|^2+\cdots+|\text{leg}|^2$$ The non-Euclidean counterparts are ... unknown!
Jul
25
revised Find the side of an equilateral triangle inscribed in a circle.
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Jul
25
revised Find the side of an equilateral triangle inscribed in a circle.
Cleaned-up formatting; minor edit