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• 2012 AMC 12B Problems/Problem 25
  ...counter-clockwise order and right angle at <math>A</math>, let <math>f(t)=\tan(\angle{CBA})</math>. What is <cmath>\prod_{t\in T} f(t)?</cmath> ...on <math>A'B'C'</math> labeled that way will give us <math>\tan CBA \cdot \tan C'B'A' = 1</math>. First we consider the reflection about the line <math>y=
  2 KB (356 words) - 17:10, 4 April 2020
• Mock Geometry AIME 2011 Problems/Problem 1
  Now we have <math> BE=BA\cdot\tan\angle EAB=1\cdot\tan30^\circ=\frac{\sqrt{3}}{3} </math>. Finally, <math> [A
  2 KB (376 words) - 22:06, 23 December 2022
• Mock Geometry AIME 2011 Problems/Problem 11
  ...nt to <math>OB,OC,</math> and arc <math>BC</math>. It is known that <math>\tan AOC=\frac{24}{7}</math>. The ratio <math>\frac{r_2} {r_1}</math> can be exp ...h>, so <math> \cos AOC=\frac{7}{25} </math>. Now, we have <math> \tan FOD=\tan\frac{AOC}{2}=\sqrt{\frac{1-\cos AOC}{1+\cos AOC}}=\sqrt{\frac{1-\frac{7}{25
  3 KB (432 words) - 14:12, 2 January 2012
• Mock Geometry AIME 2011 Problems/Problem 8
  ...t triangles <math>\Delta PO_1H, \Delta O_1S_1O_2</math>. Similarly, <math>\tan{\angle PO_2O_1}=\frac{h}{b}=\frac{52}{39}</math> using right triangles <mat
  3 KB (522 words) - 21:25, 3 January 2012
• 1989 AHSME Problems
  <math> \cot 10+\tan 5 = </math> Find the sum of the roots of <math>\tan^2x-9\tan x+1=0</math> that are between <math>x=0</math> and <math>x=2\pi</math> radi
  15 KB (2,247 words) - 13:44, 19 February 2020
• 1991 AHSME Problems/Problem 21
  \text{(C) } tan^2\theta\quad
  738 bytes (126 words) - 21:56, 17 October 2016
• 1989 AHSME Problems/Problem 28
  Find the sum of the roots of <math>\tan^2x-9\tan x+1=0</math> that are between <math>x=0</math> and <math>x=2\pi</math> radi ...</math> are positive and distinct, so by considering the graph of <math>y=\tan x</math>, the smallest two roots of the original equation <math>x_1,\ x_2</
  2 KB (282 words) - 21:14, 2 March 2019
• 1989 AHSME Problems/Problem 14
  <math>\cot 10+\tan 5=</math> We have <cmath>\cot 10 +\tan 5=\frac{\cos 10}{\sin 10}+\frac{\sin 5}{\cos 5}=\frac{\cos10\cos5+\sin10\si
  534 bytes (69 words) - 16:11, 25 February 2022
• 2012 AMC 12B Problems
  ...counter-clockwise order and right angle at <math>A</math>, let <math>f(t)=\tan(\angle{CBA})</math>. What is <cmath>\prod_{t\in T} f(t)?</cmath>
  20 KB (2,681 words) - 09:47, 29 June 2023
• 2012 AMC 12B Problems/Problem 17
  ...</math> so that <math>\cos\theta=\sin(90-\theta)=s/6</math>. Then <math>m=\tan\theta=3</math>. Substituting into <math>\left(\frac{4m^2+10}{m^2+1},\frac{6 ...</math>. Setting <math>6\sin\theta=-2\cos\theta</math>, we get that <math>\tan\theta=-1/3</math>. This means <math>-1/3</math> is the slope of line <math>
  12 KB (2,183 words) - 21:05, 23 December 2023
• 2012 AIME I Problems
  ...\angle C.</math> If <math>\frac{DE}{BE} = \frac{8}{15},</math> then <math>\tan B</math> can be written as <math>\frac{m \sqrt{p}}{n},</math> where <math>m
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 12
  ...\angle C.</math> If <math>\frac{DE}{BE} = \frac{8}{15},</math> then <math>\tan B</math> can be written as <math>\frac{m \sqrt{p}}{n},</math> where <math>m ...<math>FD = 4a\sqrt{3}</math>, and <math>FB = 11a</math>. Finally, <math>\tan{B} = \tfrac{DF}{FB}=\tfrac{4\sqrt{3}a}{11a} = \tfrac{4\sqrt{3}}{11}</math>.
  9 KB (1,523 words) - 12:23, 7 September 2022
• Mock AIME II 2012 Problems/Problem 6
  pair tan = reflect(origin,(5,y))*(0,y); draw((5,y)--tan,linetype("4 4"));
  3 KB (478 words) - 03:06, 5 April 2012
• 2006 SMT/Calculus Problems
  Evaluate: <math> \int(x\tan^{-1}x)dx </math> <cmath> \frac{d}{dx}\tanh^{-1}\tan x </cmath>
  3 KB (525 words) - 13:59, 27 May 2012
• 2006 SMT/Team Problems
  ...exist two positive numbers <math> x </math> such that <math> \sin(\arccos(\tan(\arcsin(x))))=x </math>. Find the product of the two possible <math> x </ma
  6 KB (910 words) - 17:32, 27 May 2012
• 1953 AHSME Problems/Problem 50
  ...rac{2}{3}</math>, <math>\tan{\frac{B}{2}} = \frac{1}{2}</math>, and <math>\tan{\frac{C}{2}} = \frac{4}{x},</math> so we have the equation <math>\frac{1}{2
  2 KB (314 words) - 21:17, 31 December 2023
• 2013 AMC 12B Problems
  ..., Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math> are the vertices of a trapezoid. What is <math>\sin(2x)</math>?
  16 KB (2,459 words) - 02:46, 30 January 2021
• 2013 AMC 12B Problems/Problem 20
  ..., Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math> are the vertices of a trapezoid. What is <math>\sin(2x)</math>? Let <math>f,g,h,j</math> be <math>\sin, \cos, \tan, \cot</math> (not respectively). Then we have four points <math>(f,f^2),(g,
  2 KB (375 words) - 00:54, 28 September 2021
• 1982 USAMO Problems/Problem 3
  ...} \cdot \frac{r}{s-b} \cdot \frac{r}{s-c} = \frac{1}{4} \tan A/2 \tan B/2 \tan C/2.</cmath> Lemma. <math>\tan x \tan (A - x)</math> is increasing on <math>0 < x < \frac{A}{2}</math>, where <ma
  2 KB (376 words) - 23:29, 18 May 2015
• 2013 AIME I Problems/Problem 14
  ...tter. Only the numerator, because we are trying to find <math>\frac{P}{Q}=\tan\text{arg}(\Sigma)</math> a PROPORTION of values. So denominators would canc
  10 KB (1,654 words) - 13:24, 13 June 2024

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