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• Trigonometry
  * '''Tangent''': The tangent of angle <math>A</math>, denoted <math>\tan (A)</math>, is defined as the r .../math>, denoted <math>\cot (A)</math>, is defined as the reciprocal of the tangent of <math>A</math>. <cmath>\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adj
  8 KB (1,217 words) - 20:15, 7 September 2023
• 2004 AMC 12A Problems
  ...meter <math>\overline{AB}</math> is constructed inside the square, and the tangent to the semicircle from <math>C</math> intersects side <math>\overline{AD}</ ...h> and <math>C</math> are externally tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are con
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2000 AMC 12 Problems
  ...eometry)|tangent]] to the circle at <math>A</math> and <math>\angle AOB = \theta</math>. If point <math>C</math> lies on <math>\overline{OA}</math> and <mat ...}\ \frac{1}{1+\sin\theta} \qquad \textbf {(E)}\ \frac{\sin \theta}{\cos^2 \theta}</math>
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2006 AMC 12A Problems/Problem 19
  ...h>9</math>, respectively. The equation of a common external [[tangent line|tangent]] to the circles can be written in the form <math>y=mx+b</math> with <math> ...ath> and the x-axis, so <math>m=\tan{2\theta}=\frac{2\tan\theta}{1-\tan^2{\theta}}=\frac{120}{119}</math>. We also know that <math>L_1</math> and <math>L_2<
  2 KB (253 words) - 22:52, 29 December 2021
• 2005 AIME II Problems/Problem 15
  ...rnally [[tangent (geometry)|tangent]] to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> ...etween their centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math>|r_1 - r_2|</math>. So we have
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  Let <math>\angle CAD = \angle BAE = \theta</math>. Note by Law of Sines on <math>\triangle BEA</math> we have <cmath>\frac{BE}{\sin{\theta}} = \frac{AE}{\sin{B}} = \frac{AB}{\sin{\angle BEA}}</cmath>
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 14
  ...triangle]] because <math>OB</math> is a radius and <math>BA</math> is a [[tangent line]] at point <math>B</math>. We use the [[Pythagorean Theorem]] to find ...ave that <math>\cos\theta=\frac{1}{5}</math> which implies that <math>\sin\theta=\frac{2\sqrt{6}}{5}.</math> Note that the portion of rope not on the tower
  4 KB (729 words) - 01:00, 27 November 2022
• 1995 AIME Problems
  ...The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. ...ed by faces <math>OAB</math> and <math>OBC.</math> Given that <math>\cos \theta=m+\sqrt{n},</math> where <math>m_{}</math> and <math>n_{}</math> are intege
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1983 AIME Problems/Problem 15
  ...el to <math>BC</math>. This means that <math>BC</math> is parallel to the tangent to the given circle at <math>D</math>. ...ersection. By the problem condition, however, the circle <math>P</math> is tangent to <math>BC</math> at point <math>N</math>.
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1986 AIME Problems/Problem 15
  ...math>. Then by the [[Trigonometric_identities#Angle_addition_identities | tangent angle subtraction formula]], <cmath> \tan \theta = \tan (\theta_2 - \theta_1) = \frac{\tan \theta_2 - \tan \theta_1}{1 + \ta
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1987 AIME Problems/Problem 15
  ...1}{10}</math>. Now to find <math>\tan{\theta}</math>, we find <math>\cos{2\theta}</math> using the Pythagorean Identity, and then use the tangent double angle identity. Thus, <math>\tan{\theta} = 10-3\sqrt{11}</math>. Substituting into the original sum,
  5 KB (838 words) - 18:05, 19 February 2022
• 1991 AIME Problems/Problem 12
  ...</math>, and then we found <math>AP</math>, the segment <math>OB</math> is tangent to the circles with diameters <math>AO,CO</math>. ...a} = 4\cos^3{\theta} - 3\cos{\theta}</math>, and since we have <math>\cos{\theta} = \frac {4}{5}</math>, we can solve for <math>a</math>. The rest then foll
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1996 AIME Problems/Problem 10
  ...Trigonometry#Tangent|tangent]] function is <math>180^\circ</math>, and the tangent function is [[one-to-one]] over each period of its domain. ...math>. We can set <math>\alpha (\cos{96^{\circ}}+\sin{96^{\circ}}) = \sin{\theta}</math>.Note that if we have <math>\alpha</math> equal to both the sine and
  4 KB (503 words) - 15:46, 3 August 2022
• 1999 AIME Problems/Problem 14
  ...have lengths <math>AB=13, BC=14,</math> and <math>CA=15,</math> and the [[tangent]] of angle <math>PAB</math> is <math>m/n,</math> where <math>m_{}</math> an real theta = 29.66115; /* arctan(168/295) to five decimal places .. don't know other w
  7 KB (1,184 words) - 13:25, 22 December 2022
• 2002 AIME II Problems/Problem 15
  ...The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form ...me positive reals <math>a</math> and <math>b</math>. These two circles are tangent to the <math>x</math>-axis, so the radii of the circles are <math>a</math>
  7 KB (1,182 words) - 09:56, 7 February 2022
• Leonhard Euler
  ...es that <math>e^{i\theta}=\cos(\theta)+i\sin(\theta)</math> for all <math>\theta</math>. He also discovered the power series for the [[tangent function|arctangent]], which is
  3 KB (500 words) - 21:28, 15 September 2008
• 2007 AIME I Problems/Problem 9
  ...o the extension of [[leg]] <math>CB</math>, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as ...s. As <math>\overline{AF}</math> and <math>\overline{AD}</math> are both [[tangent]]s to the circle, we see that <math>\overline{O_1A}</math> is an [[angle bi
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2007 AIME I Problems/Problem 12
  var theta=15; ...tension(A,dir(75),B/2,bisectorpoint(A,B)), Cp=rotate(theta,A)*C, Bp=rotate(theta,A)*B, X=extension(A,Bp,B,C), Y=extension(B,C,Bp,Cp);
  10 KB (1,458 words) - 20:50, 3 November 2023
• 1960 IMO Problems/Problem 6
  Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so t Now, let <math>\theta</math> be the angle subtended by a diameter of the base of the cone at the
  7 KB (1,214 words) - 18:49, 29 January 2018
• 2000 AMC 12 Problems/Problem 17
  ...</math> is tangent to the circle at <math>A</math> and <math>\angle AOB = \theta</math>. If point <math>C</math> lies on <math>\overline{OA}</math> and <mat label("$\theta$",(0.1,0.05),ENE);
  6 KB (979 words) - 12:50, 17 July 2022

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