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- ...\,</math> determine the number of times the light beam will bounce off the two line segments. Include the first reflection at <math>C\,</math> in your co ...a * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*expi(beta * pi/180);2 KB (303 words) - 00:03, 28 December 2017
- Using DeMoivre, <math>13 - t = \text{cis}\left(\frac {(2k + 1)\pi}{10}\right)</math> where <math>k</math> is an integer between <math>0</math ...pi}{10}\right) \Rightarrow \bar{t} = 13 - \text{cis}\left(-\frac {(2k + 1)\pi}{10}\right)</math>.3 KB (383 words) - 20:30, 16 June 2024
- ...d the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positi Let the center of the circle be <math>O</math>, and the two chords be <math>\overline{AB}, \overline{CD}</math> and intersecting at <ma3 KB (484 words) - 13:11, 14 January 2023
- pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D); Dividing the two equalities yields5 KB (710 words) - 21:04, 14 September 2020
- ...are not common to both equations/sets, or else we are overcounting a root two times, rather than once. Try out some equation to see where this might appl Now, we see the <math>z^2+z-1</math> and it reminds us of the sum of two cubes. Cleverly factoring, we obtain that..6 KB (1,022 words) - 20:23, 17 April 2021
- Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> ...uates to an integer ([[triangular number]]), and the [[cosine]] of <math>n\pi</math> where <math>n \in \mathbb{Z}</math> is 1 if <math>n</math> is even a1 KB (225 words) - 02:20, 16 September 2017
- Let <math>x=e^{\frac{i\pi}{36}}</math>. By Euler's Formula, <math>\sin{5k^\circ}=\frac{x^k-\frac{1}{x We factor the <math>\frac{1}{2i}</math> and split into two geometric series to get <math>\frac{1}{2i}\left(\frac{-\frac{1}{x^{35}}(x^{4 KB (614 words) - 04:38, 8 December 2023
- ...re positive, <math>z</math> lies in the first quadrant and <math>\theta < \pi/2</math>; hence by right triangle trigonometry <math>\sin \theta = \frac{\s ...point of <math>OP</math> is <math>(0.5c, 0.5d)</math>. Since the slopes of two respectively nonvertical and nonhorizontal lines have a product of <math>-16 KB (1,010 words) - 19:01, 24 May 2023
- ...the volume of the liquid can be found by <math>\frac{\pi}{3}r^2h - \frac{\pi}{3}(r')^2h'</math>. ...rac{\pi}{3}\left(\frac{3}{4}r\right)^2 \left(\frac{3}{4}h\right) &= \frac{\pi}{3}\left(r^2h - \left(\frac{rh'}{h}\right)^2h'\right)\\4 KB (677 words) - 16:33, 30 December 2023
- ...ion bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The [[ratio]] of the total area of the gre ...\ldots - 1^{2} \pi</math>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is4 KB (523 words) - 15:49, 8 March 2021
- <math>f + 4 i = (b + 2 i)\left(e^{i(2 \pi / 3)}\right) = (b + 2 i)\left(-1/2 + \frac{\sqrt{3}}{2} i\right) = -\frac{b The area of the hexagon can then be found as the sum of the areas of two congruent triangles (<math>EFA</math> and <math>BCD</math>, with height <ma9 KB (1,461 words) - 15:09, 18 August 2023
- ...e log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. Find <math>n</math>. ...king it to the existing one). Thus, <math>V=\dfrac{6^2\cdot 12\pi}{2}=216\pi</math>, so <math>n=\boxed{216}</math>.1 KB (204 words) - 17:41, 30 July 2022
- ...math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q< ...{\alpha}</math> are <math>\theta = \pi-\alpha</math>, and <math>\theta = 2\pi + \alpha</math>.2 KB (336 words) - 19:30, 24 June 2020
- ...tball game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 po ...at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?14 KB (2,059 words) - 01:17, 30 January 2024
- ...f five for $<math>2</math>. They sell all the candy bars at a price of two for $<math>1</math>. What was the profit, in dollars? ...rac{3\pi}{16} \qquad \mathrm{(D)} \frac{\pi}{4} \qquad \mathrm{(E)} \frac{\pi}{2} </math>12 KB (1,874 words) - 21:20, 23 December 2020
- ...athrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6 </cmath> ...\mathrm{(C) \ }\pi/3 \qquad \mathrm{(D) \ }2\pi/3 \qquad \mathrm{(E) \ }3/\pi/4 </cmath>14 KB (2,102 words) - 22:03, 26 October 2018
- .../math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder when <mat ...{2}+\frac{101}{100}</math> and <math>P_{2}: x=y^{2}+\frac{45}{4}</math> be two parabolas in the cartesian plane. Let <math>\mathcal{L}</math> be the commo8 KB (1,355 words) - 14:54, 21 August 2020
- ...ath> on the arc that are the farthest away from each other. Since <math>34\pi</math> is <math>1/3</math> of the circumference of a circle with radius <ma1 KB (231 words) - 18:10, 10 July 2014
- ...h that the median drawn to the hypotenuse is the [[geometric mean]] of the two legs of the triangle. ...the triangle must be <math> \frac{ \pi }{12} </math> and <math> \frac{ 5 \pi }{12}</math>, which are not difficult to construct. Q.E.D.6 KB (939 words) - 17:31, 15 July 2023
- ...ath> and <math>b</math>. Prove that the set <math>A</math> has ''exactly'' two elements. ...ngle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the midpoint of <math>BC</math>. The poin11 KB (1,779 words) - 14:57, 7 May 2012
