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  • * Intermediate is recommended for students who can expect to pass the AMC 10/12. ==== Intermediate ====
    24 KB (3,198 words) - 19:44, 4 December 2024
  • The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math> ...er]] cannot be [[negative]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <
    5 KB (860 words) - 14:36, 10 December 2023
  • '''Trigonometry''' is the study of relations between the side lengths and angles of triangl ...]]; many more, such as [[Stewart's Theorem]], are most easily proven using trigonometry. In algebra, expressions involving the trigonometric functions appear frequ
    8 KB (1,217 words) - 19:15, 7 September 2023
  • === Solution 1 (trigonometry) === === Solution 5 (Complex Numbers) ===
    13 KB (2,080 words) - 12:14, 23 July 2024
  • ...<math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity. How many distinct elements are in <math>C_{}^{}</math>? ...th>144</math>, so define <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^8, n^{16}, \ldots n^{144}\}</math> and o
    3 KB (564 words) - 03:47, 4 August 2023
  • Consider the points on the [[complex plane]]. The point <math>b+37i</math> is then a rotation of <math>60</math> Using the Pythagorean theorem with these beastly numbers doesn't seem promising. How about properties of equilateral triangles? <mat
    5 KB (787 words) - 01:09, 28 June 2024
  • In a similar fashion, we encode the angles as complex numbers, so if <math>BM=x</math>, then <math>\angle BAD=\text{Arg}(11+xi)</math> an [[Category:Intermediate Geometry Problems]]
    7 KB (1,149 words) - 12:36, 26 November 2024
  • ...1)}{(\cos 1^\circ - 1)^2 + \sin^2 1^\circ}</math> (after multiplying by [[complex conjugate]]) ...} n^\circ + \text{cis } 45-n^\circ)</math> is the sum of collinear complex numbers, so the angle the sum makes with the real axis is <math>22.5^\circ</math>.
    13 KB (1,890 words) - 21:33, 27 November 2024
  • ...s. This [[function]] has the property that the image of each point in the complex plane is [[equidistant]] from that point and the [[origin]]. Given that <m Suppose we pick an arbitrary point on the [[complex plane]], say <math>(1,1)</math>. According to the definition of <math>f(z)<
    6 KB (1,010 words) - 18:01, 24 May 2023
  • * Intermediate (at the level of hardest [[AMC 12]] problems, the [[AIME]], [[ARML]], and t ...www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#begcount Details] (Course follows a [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id
    2 KB (303 words) - 15:02, 11 July 2006
  • An olympiad-level study of [[geometry]] involves familiarity with intermediate topics to a high level, a multitude of new topics, and a highly developed p * [[Trigonometry]]
    2 KB (242 words) - 09:16, 18 June 2023
  • == Solution 3: Complex Numbers == Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, <math>\arctan\frac{1}{n}</math>, is t
    3 KB (490 words) - 21:36, 28 November 2023
  • Let <math>a</math> and <math>b</math> be positive real numbers with <math>a\ge b</math>. Let <math>\rho</math> be the maximum possible val ...>, so <math>(x,b)</math> is the top vertex. Now we can compute (by complex numbers, or the sine angle addition identity) that <math>b = \frac{\sqrt{3}}{2}a +
    8 KB (1,387 words) - 10:56, 29 January 2024
  • To save us from getting big numbers with lots of zeros behind them, let's divide all side lengths by <math>200< ==Solution 4 (Trigonometry Bash)==
    25 KB (3,940 words) - 18:48, 24 October 2024
  • The solutions to the equation <math>(z+6)^8=81</math> are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled == Solution 1 (Complex Numbers in Polar Form) ==
    11 KB (1,708 words) - 11:01, 18 March 2023
  • ==Solution 1 (Complex Numbers: Vieta's Formulas)== ==Solution 2 (Complex Numbers: Trigonometric Identities)==
    10 KB (1,507 words) - 12:52, 21 November 2024
  • Trigonometry Intermediate Algebra
    1,005 bytes (92 words) - 22:11, 17 January 2021