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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
• Isogonal conjugate
  Let <math>\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.</math> ...= \frac {PC \sin \theta}{PC \sin \Theta} = \frac {QC \sin \theta}{QC \sin \Theta} = \frac {QD'}{QE'}. \blacksquare</cmath>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 12A Problems/Problem 20
  label("\(\sin \theta = \frac{3}{5}\)",B-(.2,-.1),W); ...<math>CD</math> tangent to <math>O_a</math> <math>M</math>, and the point tangent to <math>O_b</math> <math>N</math>. Since <math>\triangle CO_aM</math> and
  6 KB (951 words) - 16:31, 2 August 2019
• 2008 AMC 12A Problems/Problem 24
  label("\(\theta\)",(7,.4)); Let <math>x = CA</math>. Then <math>\tan\theta = \tan(\angle BAF - \angle DAE)</math>, and since <math>\tan\angle BAF = \f
  3 KB (513 words) - 14:35, 7 June 2018
• 2008 AIME I Problems/Problem 14
  ...<math>T</math> lies on <math>\omega</math> so that line <math>CT</math> is tangent to <math>\omega</math>. Point <math>P</math> is the foot of the perpendicul label("\(\theta\)",C + (-1.7,-0.2), NW);
  8 KB (1,333 words) - 00:18, 1 February 2024
• 2008 AIME II Problems/Problem 11
  .... Circle <math>Q</math> is externally [[tangent]] to <math>P</math> and is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math>. No point of c ...t <math>\angle ACB = 2\theta</math>; then <math>\angle PCX = \angle QBX = \theta</math>. Dropping the altitude from <math>A</math> to <math>BC</math>, we re
  6 KB (1,065 words) - 20:12, 9 August 2022
• 1998 AHSME Problems/Problem 28
  ...nd <math>3 \theta = \angle CAB</math>. Then, it is given that <math>\cos 2\theta = \frac{AC}{AD} = \frac{2}{3}</math> and ...tan 3\theta - \tan 2\theta)}{AC \tan 2\theta} = \frac{\tan 3\theta}{\tan 2\theta} - 1.</math></center>
  4 KB (662 words) - 00:51, 3 October 2023
• 2008 iTest Problems
  ...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 ...ists a circle, lying inside the quadrilateral and having center I, that is tangent to all four sides of the quadrilateral. Points <math>M</math> and <math>N</
  71 KB (11,749 words) - 01:31, 2 November 2023
• Spiral similarity
  ...ath> is any point of <math>AB, \theta </math> is circle <math>CPx_0, C' = \theta \cap A'B'</math> is the image <math>C</math> under spiral symilarity center ...x_0 = \Omega \cap \omega, x_0 \neq B, C </math> is any point of <math>AB, \theta </math> is circle <math>CBx_0,</math>
  28 KB (4,863 words) - 00:29, 16 December 2023
• 1992 USAMO Problems/Problem 4
  ...ough <math>A</math>, <math>B</math>, <math>C</math>, and <math>P</math> is tangent to the sphere through <math>A'</math>, <math>B'</math>, <math>C'</math>, an ...e that pass through <math>P</math> for the <math>2</math> spheres that are tangent to each other.
  5 KB (807 words) - 18:37, 25 June 2021
• 2009 AMC 10B Problems/Problem 20
  ...\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 - BD^2} Remarks: You could also use tangent half angle formula
  2 KB (371 words) - 15:34, 15 October 2023
• 2009 AIME I Problems/Problem 12
  ...ch that <math>\overline{AI}</math> and <math>\overline{BI}</math> are both tangent to circle <math>\omega</math>. The ratio of the perimeter of <math>\triangl ...}}</math>. By the double-angle formula <math>\sin(2\theta)=2\sin\theta\cos\theta</math>, it turns out that <cmath>\sin(\angle BAI)=\sin(2\angle DAO)=\dfrac{
  12 KB (1,970 words) - 22:53, 22 January 2024
• 2010 AMC 12B Problems/Problem 18
  ...le, which is itself centered on <math>(1,0)</math>. Let us define <math>f(\theta)</math> as the value of the length of the first circle that lies within the <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath>
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2011 AMC 12A Problems/Problem 24
  ...ath>\angle ABC=180^{\circ}-\alpha</math> and <math>\angle BCD=180^{\circ}-\theta</math>. Let the circle have center <math>O</math> and radius <math>r</math> ...\theta</math>, <math>GOH=180^{\circ}-\alpha</math>, <math>EOH=180^{\circ}-\theta</math>, and <math>FOE=\alpha</math>.
  4 KB (753 words) - 18:58, 2 June 2022
• 2011 USAJMO Problems/Problem 5
  ...ven points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> Since PB and PD are tangent to the circle, it's easy to see that M is the midpoint of arc BD.
  4 KB (717 words) - 17:00, 14 April 2024
• 2011 USAJMO Problems/Problem 3
  ...nnot be on <math>p</math>. This implies that <math>q</math> is exactly the tangent line to <math>p</math> at <math>P</math>, that is <math>q=\ell(P)</math>. S ...h> be the reflection of <math>F</math> across <math>q</math>. Then <math>2\theta=\angle FBH=\angle C'HB</math>, and so <math>\angle C'HB=\angle AHB</math>.
  15 KB (2,593 words) - 13:37, 29 January 2021
• 1984 AHSME Problems/Problem 29
  ...math>OA=\sqrt{18}</math>, <math>\angle OXA = 90</math> because the line is tangent to the circle. Using the pythagorean theorem, we have <math>OX=\sqrt{12}</m ...th>\tan (\theta + 45)</math> where <math>\angle AOX = \theta</math>. Using tangent addition,
  4 KB (614 words) - 20:09, 12 September 2022
• 1979 USAMO Problems/Problem 2
  ...the equator, then <math>C=(cos(\theta),sin(\theta),0)</math> where <math>\theta</math> is the angle on the <math>xy</math>-plane from the origin to <math>C ...CN}}</math> is the unit vector in the direction of arc <math>CN</math> and tangent to the great circle of <math>CN</math> at <math>C</math>
  6 KB (1,013 words) - 22:09, 21 November 2023
• 1996 AHSME Problems/Problem 25
  ...misses the circle altogether. This means <math>3x + 4y = A</math> will be tangent to the circle. ...will be <math>\frac{4}{3}</math>, since the radius is perpendicular to the tangent line.
  9 KB (1,441 words) - 17:51, 22 October 2023
• Barycentric coordinates
  The incircle contains the tangent points of the incircle with the sides: Denote <math>R = PQ \cap P'Q', \theta = \odot P'QR, F = \Omega \cap \theta \notin \odot PQ'R, D \in \Omega</math> is the point isogonal conjugate to l
  25 KB (5,067 words) - 22:15, 31 March 2024
• Mathtime Version 1 Issue 2
  ...e has a circle inscribed within the circle so that the inscribed circle is tangent to all <math>3</math> sides. What is the ratio of the sum of the areas of a ...ouches the origin and has its center located on the y-axis. The circle is tangent to the line <math>x + 3y = 6</math>. Given that the radius of the circle c
  15 KB (2,444 words) - 21:46, 1 January 2012
• 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>
  15 KB (2,549 words) - 08:36, 2 September 2023
• 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

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