The mathematician formerly known as DayLateDon

13h
comment What is the smallest possible angle of this polygon?
Why isn't the smallest-area polygon simply the small square itself (area $1$)?
17h
comment What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$?
Help us help you. Explain what you've tried, and where you got stuck. This will help others provide answers targeted to your skill level.
17h
comment External angle bisectors of a triangle
@DanUznanski: Internal angle bisector lines pass through the interior of the triangle; exterior angle bisector lines ---that is, lines bisecting the *exterior angles*--- do not. The interior bisector at a vertex is in fact perpendicular to the external bisector at that vertex. Definitely not the same line.
17h
comment External angle bisectors of a triangle
The angle bisectors are probably to be thought of as lines through $B$ and $C$, not rays.
19h
comment Non-trigonometric Continuous Periodic Functions
You could chain together copies of the bumpy part of the typical Bump function.
20h
comment If α, β are two values of θ satisfying equation cosθ/a + sinθ/b = 1/c then prove that cot ((α+β)/2) = b/a
Related (but not duplicate): "Cosine of the sum of two solutions of trigonometric equation $a\cos\theta+b\sin\theta=c$". Now, my answer to that question involves a diagram that clearly shows (in the notation used there) $\cot((\theta+\phi)/2) = a/b$. Replacing that question's $a$ and $b$ (and $c$) with your question's $1/a$ and $1/b$ (and $1/c$) gives the result you seek. Other answers to that question provide additional approaches.
21h
answered Prove concurrency (probably using Carnot's Theorem)
1d
comment How to isolate x in this case?
You'll notice that you've arrived back at your original equation! :) Try combining $u^3 + v^3 = - q$ with the fact that $p^3 = - 27 u^3 v^3$. (To reduce exponent clutter, you can think of this as $U+V=-q$ and $p^3 = -27 UV$. Solve for $U$ and $V$.)
1d
comment Prove concurrency (probably using Carnot's Theorem)
Use the "Power of a Point with respect to a Circle" to relate various segments lengths. For instance, $$|\overline{AA_c}||\overline{AB}| = \text{power of $A$ wrt $K_a$} = |\overline{AA_b}||\overline{AC}|$$ Also, $|\overline{AA_c}| + |\overline{BA_c}| = |\overline{AB}|$, etc; and $|\overline{AC_c}| = \frac12(|\overline{AB_c}| + |\overline{AA_c}|)$, etc, where $C_c$ is the midpoint of $\overline{A_cB_c}$.
1d
comment Viviani on Sphere parametrization
For reference: Viviani's Curve
2d
comment Proving the trigonometric identity with angles in GP
possible duplicate of Trigo Problem : Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$
2d
comment Geometric locus
Of course, if all you need to know is that the locus is a circle (without regard for the actual size or location of that circle), then you can stop after your rearrangement step. Wrestling the equation into the Standard Form for a Circle is overkill (although instructive). It's enough to know that, since the coefficients of $x^2$ and $y^2$ match (and there's no $xy$ term), you could wrestle the equation into the Standard Form.
2d
comment Geometric locus
Writing $P=(x,y)$, if you were to expand everything (which you don't really have to do), you'd get an equation that's quadratic in $x$ and $y$, with no $xy$ term. Moreover, the coefficients on $x^2$ and $y^2$ would match. Therefore, ...
2d
answered Prove that secants of a circle pass through a common point
Sep
15
comment Are addition and multiplication on naturals algebraically distinguishable?
@user107952: Good to know! :) I've converted my comment to an official answer. Please accept it to remove this question from the "Unanswered" queue.
Sep
15
answered Are addition and multiplication on naturals algebraically distinguishable?
Sep
15
comment Are addition and multiplication on naturals algebraically distinguishable?
@user107952: By commutativity and associativity of addition, any expression in an "equational identity" on $(\mathbf{N},+)$ with variables $x_1$, $x_2$, $\dots$, $x_n$ (with repetition) can be re-written in the "standard form" $$(x_1+x_1+\cdots+x_1)+(x_2+x_2+\cdots+x_2)+\cdots+(x_n+x_n+\cdots+x_n)$$ where each grouping contains all copies of the corresponding variable, $x_i$. Clearly, two expressions are equal if and only if they have the same "standard form"; therefore, the expressions have the same variables with the same number of repetitions. The same is true for $(\mathbf{N},\times)$.
Sep
15
comment Are addition and multiplication on naturals algebraically distinguishable?
@user107952: "If the answer is yes to both questions, my question is answered in the affirmative." That's my point. :) Well, and I want to verify what you mean by "equational identity". You say "we can't mix up addition and multiplication"; does that mean that the only operation allowed in equational identity on $(\mathbf{N},+)$ is addition? If so, then an expression in equational identity would seem to consist of a collection of variables (allowing repetition) separated by "$+$"s, and possibly grouped by "$()$"s ... wouldn't it? Do you intend "equational identity" to mean something broader?
Sep
14
comment Are addition and multiplication on naturals algebraically distinguishable?
Can you given an example of an equational identity for $(\mathbf{N},+)$ that is not simply an application of the commutative and associative properties of addition?
Sep
14
comment Trigonometry question from "Quick Calculus" book
You might find this answer to a more-general question helpful.
1 2 3 4 5