User Emmanuel José García - Mathematics Stack Exchange

13 votes

1 answer

6k views

The theoretical importance of the half-angle formulas

asked May 22, 2022 at 19:01

11 votes

4 answers

1k views

Trigonometric formula for solving quadratic equations

asked Jan 13 at 19:50

9 votes

1 answer

1k views

Generalization of the law of tangents for a cyclic quadrilateral

asked Mar 16, 2022 at 19:07

8 votes

2 answers

326 views

Showing $\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}} (x^2 - x) \, dx = \phi\pi$

asked Mar 28 at 5:34

5 votes

1 answer

801 views

Generalizing Lami's theorem

asked Jul 10, 2022 at 21:51

5 votes

0 answers

704 views

Generalization of two formulae and an alternative proof of Bretschneider's formula

asked Aug 22, 2020 at 15:12

4 votes

4 answers

709 views

A new regular polyhedron?

asked Apr 29, 2017 at 17:27

4 votes

3 answers

535 views

A curious family of integrals that give $\pi$

asked Jul 16, 2023 at 2:30

4 votes

0 answers

191 views

Who is the author of this integration technique?

asked Apr 1 at 17:03

4 votes

2 answers

314 views

Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

asked Jul 13 at 16:22

3 votes

1 answer

639 views

Evaluating $\int_{2}^{5}\sqrt{\tan{\left(\frac{\csc^{-1}(x)}{2}\right)}}\,dx$ using a new (?) trick

asked Feb 3 at 15:31

3 votes

0 answers

124 views

Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

asked Apr 10 at 11:43

3 votes

1 answer

134 views

Introducing a sine half-angle substitution

asked Jul 9, 2023 at 21:54

3 votes

1 answer

382 views

Equilibrium problem with 3 unknown forces | Hanging mass

asked Jul 16, 2022 at 16:42

3 votes

1 answer

481 views

On a constant associated to equilateral triangle and its generalization.

asked Jul 28, 2016 at 3:45

3 votes

0 answers

137 views

Is this proof of the Pythagorean theorem known?

asked May 9, 2020 at 17:01

3 votes

1 answer

562 views

The foot of the height in an orthocentric tetrahedron is the orthocenter of a face.

asked Jan 22, 2017 at 21:50

2 votes

0 answers

187 views

A new(?) proof of the Law of Cosines using segments determined by the points of contact with a triangle's incircle

asked Jun 6, 2020 at 2:21

2 votes

1 answer

134 views

Showing $\sum_{cyc}\tan\frac\alpha2\tan\frac\beta2\geq4$ for a cyclic quadrilateral

asked Jun 20, 2022 at 2:30

2 votes

1 answer

524 views

Sum of the Volumes of Two Tetrahedra Obtained by Reflection Equals the Volume of the Reference Tetrahedron

asked Jul 25, 2023 at 6:07

2 votes

1 answer

179 views

On Bretschneider's generalization of Ptolemy's theorem

asked Oct 4, 2022 at 11:32

2 votes

1 answer

132 views

Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$

asked Jul 4 at 16:46

1 vote

1 answer

156 views

Evaluating $\int_1^2 \frac{1}{\tan{\left(\frac{\sec^{-1}(x)}{2}\right)}-x}\,dx$

asked Mar 18 at 0:54

1 vote

1 answer

119 views

Trigonometric version of Schur's inequality

asked Dec 24, 2023 at 16:44

1 vote

0 answers

160 views

Another proof of Euler's inequality via the half-angle formulas

asked Jun 25, 2022 at 17:36

1 vote

1 answer

224 views

Yet another constant associated to regular polygons!

asked Aug 12, 2016 at 3:38

1 vote

0 answers

171 views

Still very Bretschneider, isn't it?

asked Nov 21, 2020 at 14:30

1 vote

0 answers

245 views

Generalization of the Law of Sines

asked Dec 28, 2021 at 15:58

1 vote

1 answer

548 views

An analog of Mollweide's Formula for a cyclic quadrilateral

asked Jan 10, 2022 at 17:23

1 vote

0 answers

254 views

Cosine of the sum of two angles from the half angle formulas

asked Apr 26, 2022 at 18:01

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41 Questions

The theoretical importance of the half-angle formulas

Trigonometric formula for solving quadratic equations

Generalization of the law of tangents for a cyclic quadrilateral

Showing $\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}} (x^2 - x) \, dx = \phi\pi$

Generalizing Lami's theorem

Generalization of two formulae and an alternative proof of Bretschneider's formula

A new regular polyhedron?

A curious family of integrals that give $\pi$

Who is the author of this integration technique?

Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

Evaluating $\int_{2}^{5}\sqrt{\tan{\left(\frac{\csc^{-1}(x)}{2}\right)}}\,dx$ using a new (?) trick

Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

Introducing a sine half-angle substitution

Equilibrium problem with 3 unknown forces | Hanging mass

On a constant associated to equilateral triangle and its generalization.

Is this proof of the Pythagorean theorem known?

The foot of the height in an orthocentric tetrahedron is the orthocenter of a face.

A new(?) proof of the Law of Cosines using segments determined by the points of contact with a triangle's incircle

Showing $\sum_{cyc}\tan\frac\alpha2\tan\frac\beta2\geq4$ for a cyclic quadrilateral

Sum of the Volumes of Two Tetrahedra Obtained by Reflection Equals the Volume of the Reference Tetrahedron

On Bretschneider's generalization of Ptolemy's theorem

Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$

Evaluating $\int_1^2 \frac{1}{\tan{\left(\frac{\sec^{-1}(x)}{2}\right)}-x}\,dx$

Trigonometric version of Schur's inequality

Another proof of Euler's inequality via the half-angle formulas

Yet another constant associated to regular polygons!

Still very Bretschneider, isn't it?

Generalization of the Law of Sines

An analog of Mollweide's Formula for a cyclic quadrilateral

Cosine of the sum of two angles from the half angle formulas