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• 2012 AMC 12A Problems
  ...e the acute angle formed by that side and the median. What is <math>\sin{\theta}</math>? A square region <math>ABCD</math> is externally tangent to the circle with equation <math>x^2+y^2=1</math> at the point <math>(0,1)
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2006 SMT/Team Problems
  ...theta </math> with the positive <math> x </math> axis. Compute <math> \cos\theta </math>.
  6 KB (910 words) - 17:32, 27 May 2012
• 1991 AHSME Problems
  <math>\textbf{(A) } \sin^2\theta\qquad \textbf{(B) } \cos^2\theta\qquad
  16 KB (2,451 words) - 04:27, 6 September 2021
• Symmetry
  We construct a tangent to the the circle <math>\odot DEF,</math> at the point <math>G.</math> It i ...math> and <math>E.</math> Line <math>\ell</math> is the tangent for <math>\theta</math> at the point <math>I.</math>
  19 KB (3,292 words) - 13:04, 13 May 2024
• 1981 AHSME Problems
  ...n\theta\qquad\textbf{(E)}\ \dfrac{5}{2}-\dfrac{1}{2}\sin\left(\dfrac{1}{2}\theta\right) </math> ...ond circle is tangent internally to the circumcircle at <math>T</math> and tangent to sides <math>AB</math> and <math>AC</math> at points <math>P</math> and <
  17 KB (2,633 words) - 15:44, 16 September 2023
• 2014 AIME I Problems
  ...\tfrac{w-z}{z}\right) </math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> an ...radius <math>5</math>. Let <math>A</math> be the point where the disks are tangent, <math>C</math> be the center of the smaller disk, and <math>E</math> be th
  9 KB (1,472 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 7
  ...\tfrac{w-z}{z}\right) </math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> an We know that <math>\tan{\theta}</math> is equal to the imaginary part of the above expression divided by t
  5 KB (782 words) - 20:25, 10 October 2023
• 1990 AHSME Problems
  ...,C</math> and <math>D</math>. If <math>AB=1</math> and <math>\angle{APB}=2\theta</math>, then the volume of the pyramid is <math>\textbf{(A) } \frac{\sin(\theta)}{6}\qquad
  14 KB (2,099 words) - 01:15, 10 September 2021
• 1986 AHSME Problems
  In the configuration below, <math>\theta</math> is measured in radians, <math>C</math> is the center of the circle, ...h>BCD</math> and <math>ACE</math> are line segments and <math>AB</math> is tangent to the circle at <math>A</math>.
  17 KB (2,512 words) - 18:30, 12 October 2023
• 1976 AHSME Problems
  ...n acute angle, and <math>\sin 2\theta=a</math>, then <math>\sin\theta+\cos\theta</math> equals In the adjoining figure, <math>AB</math> is tangent at <math>A</math> to the circle with center <math>O</math>; point <math>D</
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1986 AHSME Problems/Problem 21
  In the configuration below, <math>\theta</math> is measured in radians, <math>C</math> is the center of the circle, ...h>BCD</math> and <math>ACE</math> are line segments and <math>AB</math> is tangent to the circle at <math>A</math>.
  2 KB (301 words) - 18:50, 1 April 2018
• 2015 AIME II Problems/Problem 13
  ...rc is the argument of <math>z</math>), because the <math>y</math>- axis is tangent to the circle at the origin. So <math>\text{arg}(z-1)=\frac{\pi+1}{2}</math We want the argument of the whole expression <math>-\pi<\theta<0</math>. This translates into <math>\frac{-\pi-1}{2}<\text{arg}\left(z^n-1
  6 KB (1,034 words) - 21:29, 14 January 2024
• 2015 AIME II Problems/Problem 15
  ...<math>\mathcal{Q}</math> so that line <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> interse ...be the intersection of <math>\overline{BC}</math> and the common internal tangent of <math>\mathcal P</math> and <math>\mathcal Q.</math> We claim that <math
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems/Problem 15
  ...ect at points <math>X</math> and <math>Y</math>. Line <math>\ell</math> is tangent to <math>\omega_1</math> and <math>\omega_2</math> at <math>A</math> and <m pair a=tangent(q,o1,R1,2);
  14 KB (2,427 words) - 17:12, 8 January 2024
• 1979 AHSME Problems/Problem 20
  ...n(\arctan(x)) = \tan(\theta)</math>, and simplifying gives <math>x = \tan(\theta)</math>. So <math>a = \tan(\theta_a) = \frac{1}{2}</math> and <math>b = \ta ...given two tangent measures, it is natural for us to think about the sum of tangent measures (what else can we try? Remember: we are not allowed to use calcula
  2 KB (363 words) - 12:46, 10 May 2022
• MIE 2015
  ...er such that <math>\frac{2Z}{\overline{Z}i}</math> has an argument <math>(\theta)</math> equal to <math>\frac{3\pi}{4}</math> and <math>\log_3(2Z+2\overline ...ning of the movement of <math>C'</math> the point <math>P</math> is at the tangent point <math>(4,0)</math>, like in the figure a. After some movement, the an
  7 KB (1,127 words) - 18:23, 11 January 2018
• 2018 AMC 12B Problems
  ...mcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> and <math>BC</math> and to the circumcircle of ...that <math>P_1P_2=P_2P_3=P_3P_1</math> and line <math>P_iP_{i+1}</math> is tangent to <math>\omega_i</math> for each <math>i=1,2,3</math>, where <math>P_4 = P
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 12B Problems/Problem 25
  ...that <math>P_1P_2=P_2P_3=P_3P_1</math> and line <math>P_iP_{i+1}</math> is tangent to <math>\omega_i</math> for each <math>i=1,2,3</math>, where <math>P_4 = P real theta = 41.5;
  13 KB (2,080 words) - 19:09, 21 October 2023
• 1973 AHSME Problems/Problem 17
  ...ath> \sin \frac12 \theta = \sqrt{\frac{x-1}{2x}}</math>, then <math> \tan \theta</math> equals .../math> is acute and <math>\cos \tfrac{\theta}{2} = \sqrt{1 - \sin (\tfrac{\theta}{2})^2}</math>,
  1 KB (184 words) - 14:00, 20 February 2020
• 2008 iTest Problems/Problem 57
  ...sible values of <math>\tan\theta</math> given that <math>\sin\theta + \cos\theta = \dfrac{193}{137}</math>. If <math>a+b=m/n</math>, where <math>m</math> an <cmath>\frac{\sqrt{2}}{2}\sin\theta + \frac{\sqrt{2}}{2}\cos\theta = \frac{\sqrt{2}}{2}x</cmath>
  2 KB (343 words) - 20:35, 4 August 2018

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