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• 2009 AMC 10A Problems/Problem 10
  ...get <math>xy = 3</math>, and using the fact that <math>\cot{a} = \frac{1}{\tan{a}}</math>, <math>\frac{x}{y} = 4</math>. Solving for x and y(using substit
  5 KB (879 words) - 18:57, 30 April 2024
• 2001 AMC 12 Problems/Problem 24
  ...<math>\angle ACH</math> can be simplified. Indeed, if you know that <math>\tan(75)=2+\sqrt{3}</math> or even take a minute or two to work out the sine and ...= 2 + \sqrt{3}</math>. Looking that the answer options we see that <math>\tan{75^\circ} = 2 + \sqrt{3}</math>. This means the answer is <math>D</math>.
  8 KB (1,316 words) - 22:48, 7 March 2024
• 2007 AMC 12B Problems/Problem 22
  ...}</math> and the <math>x</math>-axis is <math>30^{\circ}</math>, and <math>tan(30) = \frac{\sqrt{3}}{3}</math>.
  4 KB (707 words) - 16:36, 15 February 2021
• 2009 AMC 10B Problems/Problem 20
  ...e BAD = \angle DAC</math>. Notice <math>\tan \theta = BD</math> and <math>\tan 2 \theta = 2</math>. By the double angle identity, <cmath>2 = \frac{2 BD}{1
  2 KB (371 words) - 15:34, 15 October 2023
• 2009 AIME I Problems/Problem 12
  ...t <math>O</math> be the center of <math>\omega</math>; notice that <cmath>\tan(\angle DAO)=\dfrac{DO}{AD}=\dfrac{210/37}{144/37}=\dfrac{35}{24}</cmath> so Since we have <math>\tan OAB = \frac {35}{24}</math> and <math>\tan OBA = \frac{6}{35}</math> , we have <math>\sin {(OAB + OBA)} = \frac {1369}
  12 KB (1,970 words) - 22:53, 22 January 2024
• 2009 AIME I Problems/Problem 15
  ...BOP</math> and <math>COP</math>, with <math>BO=CO=7</math> and <math>OP=7 \tan 15=7(2-\sqrt{3})=14-7\sqrt3</math>. Then, the area of [<math>\triangle BPC<
  6 KB (1,048 words) - 19:35, 2 January 2023
• 2009 AIME II Problems/Problem 15
  <cmath>\frac{NV}{MV} = \frac{\sin (\alpha)}{\sin (90^\circ - \alpha)} = \tan (\alpha)</cmath> ...math>VW = NW + MV - 1 = \frac{1}{1+\frac{3}{4}\cot(\alpha)} + \frac{1}{1+\tan (\alpha)} - 1</math>. Taking the derivative of <math>VW</math> with respect
  11 KB (1,862 words) - 21:23, 23 May 2024
• Alternating permutation
  \sum_{n = 0}^\infty \frac{E_n}{n!} x^n = \sec x + \tan x .
  2 KB (246 words) - 12:50, 6 August 2009
• 1998 USAMO Problems/Problem 3
  ...rac {\pi}{4}\right)} + \tan{\left(a_1 - \frac {\pi}{4}\right)} + \cdots + \tan{\left(a_n - \frac {\pi}{4}\right)}\ge n - 1</cmath> Prove that <math>\tan{\left(a_0\right)}\tan{\left(a_1\right)}\cdots \tan{\left(a_n\right)}\ge n^{n + 1}</math>.
  2 KB (322 words) - 13:31, 23 August 2023
• Differentiation Rules
  If <math>y(x) = \tan x</math>, then <math>\frac{dy}{dx} = \sec^2 x</math>. Note that this follow
  2 KB (309 words) - 10:50, 4 June 2024
• 2010 AMC 12A Problems/Problem 8
  <cmath>\frac{\sqrt{3}}{2}\cos(x)=\frac{3}{2}\sin(x)\implies \tan(x)=\frac{\sqrt{3}}{3}</cmath>
  4 KB (621 words) - 15:12, 21 June 2023
• 2010 AMC 12B Problems
  ...ath> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, and <math>a_3= \tan x</math> for some real number <math>x</math>. For what value of <math>n</ma
  12 KB (1,845 words) - 13:00, 19 February 2020
• Mock AIME 1 2010 Problems
  ...sqrt{159}}{7}</math>, and <math>ED^2 + EB^2 = 3050</math>, and that <math>\tan m \angle ACB</math> can be expressed in the form <math>\frac{a \sqrt{b}}{c}
  7 KB (1,297 words) - 01:29, 25 November 2016
• 2010 AIME I Problems/Problem 15
  <cmath>r = AX \tan(A/2) = CY \tan(C/2)</cmath> ...^2}{2\cdot 12\cdot 15} = \frac{200}{30\cdot 12}=\frac{5}{9}</math>, <math>\tan(A/2) = \sqrt{\frac{1-\cos A}{1+\cos A}} = \sqrt{\frac{9-5}{9+5}} = \frac{2}
  14 KB (2,210 words) - 13:14, 11 January 2024
• Mock AIME 2 2010 Answer Key
  Hence, <math>(4 \sin^2(x+y) - 7 \cos^2(x+y)) \leq 0</math>, yielding <math>\tan^2(x+y) = \frac{7}{4}</math>, or <math>x + y = \pm \arctan{\frac{\sqrt{7}}{2
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 USAMO Problems/Problem 1
  <math>AY \cos(\delta)\sin(\alpha)\tan(\delta) = AY \sin(\delta)\sin(\alpha)</math>
  13 KB (2,271 words) - 14:08, 23 May 2024
• 1966 IMO Problems
  ...se sides. Prove that if <cmath> a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) </cmath> the triangle is isosceles.
  3 KB (509 words) - 12:39, 29 January 2021
• 2010 AMC 12B Problems/Problem 20
  ...ath> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, and <math>a_3= \tan x</math> for some real number <math>x</math>. For what value of <math>n</ma ...ometric sequence, we have <math>\cos^2x=\sin x \tan x</math>. Since <math>\tan x=\frac{\sin x}{\cos x}</math>, we can rewrite this as <math>\cos^3x=\sin^2
  2 KB (375 words) - 19:52, 24 June 2022
• Mock USAJMO
  | Tan
  3 KB (295 words) - 20:01, 4 March 2020
• 2011 AMC 12A Problems/Problem 24
  ...\frac{C}{2}=90^\circ-\angle \frac{A}{2}</math>, so <math>\cot\frac{C}{2}=\tan\frac{A}{2}</math>, and we find that <math>\frac{r}{x}=\frac{x-5}{r}</math>. ...rac{B}{2}=\frac{14-x}{r}</math>, and <math>\cot\frac{D}{2}=\frac{12-x}{r}=\tan\frac{B}{2}=\frac{r}{14-x}</math>. Therefore, <math>r^2=\frac{12-x}{14-x}</m
  4 KB (753 words) - 18:58, 2 June 2022

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