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• Trigonometric identities
  The plus or minus does not mean that there are two answers, but that the sign of the expression depend These are the identities that are not substantial enough to warrant a section of their own.
  8 KB (1,397 words) - 20:55, 20 January 2024
• 1983 AIME Problems/Problem 4
  ==Solution 2 (Trig)== ==Solution 4 (Trig)==
  11 KB (1,741 words) - 21:40, 23 November 2023
• 1983 AIME Problems/Problem 14
  ...sume that <math>R</math> lies on the line connecting the centers, which is not true in general. It is true here only because the perpendicular from <math> === Solution 3 (trig bash)===
  13 KB (2,174 words) - 15:52, 10 November 2024
• 1983 AIME Problems/Problem 15
  ...intersect the chord <math>BC</math> at two points, one point, or they may not have a point of intersection. By the problem condition, however, the circle ...ave found one possible case. However, in a full-solution contest, we could not assume that the answer is unique, but would need to prove that this is the
  20 KB (3,497 words) - 14:37, 27 May 2024
• 1986 AIME Problems/Problem 3
  == Solution 3 (less trig required, use of quadratic formula) == Remark: The quadratic need not be solved. The value of <math>ab</math> can be found through Vieta's.
  3 KB (545 words) - 22:44, 12 October 2023
• 1997 AIME Problems/Problem 15
  ...>q</math>, and <math>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ==Solution 4 (Fast, no trig)==
  5 KB (811 words) - 20:39, 20 July 2024
• 2001 AIME I Problems/Problem 4
  ...>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. ==Solution 2 (no trig)==
  4 KB (695 words) - 09:02, 9 October 2024
• 2003 AIME I Problems/Problem 11
  ...rs <math> \sin^2 x, \cos^2 x, </math> and <math> \sin x \cos x </math> are not the lengths of the sides of a triangle. Given that <math> p = d/n, </math> ...ath> is the largest of the three given expressions and those three lengths not forming a [[triangle]] is equivalent to a violation of the [[triangle inequ
  5 KB (895 words) - 14:24, 21 June 2024
• 2003 AIME II Problems/Problem 14
  ...th>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n.</math> ==Solution 4 (No Trig)==
  9 KB (1,463 words) - 17:08, 31 July 2024
• 2005 AMC 10A Problems/Problem 25
  ==Solution 3 (trig)== Note: <math>BC=39</math> was not used in this problem.
  6 KB (897 words) - 13:31, 13 October 2024
• 2006 AIME II Problems/Problem 12
  ...th> and <math> r</math> are [[relatively prime]], and <math> q </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any prim == Solution 3: Trig ==
  6 KB (1,033 words) - 01:36, 19 March 2022
• 2008 AMC 12B Problems/Problem 25
  Note: Once <math>DY</math> is found, there is no need to do the trig. Notice that the hexagon consists of two trapezoids, <math>ABPQ</math> and ...= X, AP \cap CD = Y.</math> Then, angle chasing shows that <math>CQ</math> not only bisects <math>BX,</math> but is also perpendicular to it. This makes i
  13 KB (2,244 words) - 21:31, 13 October 2024
• 2009 AMC 10A Problems/Problem 10
  ...very improbably due to the area being <math>\frac{7x}{2}</math>. This does not work. ...\frac{3}{x}</math>, and <math>\tan{90-\theta} = \frac{4}{x}</math>. Using trig identities, we can simplify the second one to <math>\cot{\theta} = \frac{4}
  5 KB (886 words) - 09:38, 16 November 2024
• 2001 AMC 12 Problems/Problem 24
  ...irc</math> and <math>\angle ADC = 60^\circ</math> as you can see, let's do trig. Drop an altitude from <math>A</math> to <math>BC</math>; call this point < ...75^\circ</math> or <math>105^\circ</math>. Since <math>105^\circ</math> is not a choice, we know <math>\angle ACB = \boxed{75^\circ}</math>.
  8 KB (1,316 words) - 21:48, 7 March 2024
• 2009 AMC 12B Problems/Problem 24
  ...th>\sin^{-1}(\sin(6x)) = 180 - 6x</math>. As <math>\frac{180}{7}</math> is not on the interval <math>30 \leq x \leq 45</math>, this yields no solution. .... Thus, <math>\sin^{-1}(\sin(6x)) = 6x - 360</math>. As <math>72</math> is not on the interval <math>45 \leq x \leq 60</math>, this yields no solution.
  7 KB (1,287 words) - 06:09, 22 December 2022
• 2009 AIME I Problems/Problem 12
  (If it's not then the answer to the problem would be irrational which can't be in the fo ...can know that <math>a</math> has to be a multiple of <math>1369</math> and not of <math>3</math> and it's reasonable to think that <math>b</math> is divis
  13 KB (2,170 words) - 22:35, 3 September 2024
• 2010 AMC 12A Problems/Problem 13
  ...k</math> do the graphs of <math>x^2+y^2=k^2</math> and <math>xy = k</math> not intersect? ...h of the circle. This should be enough to conclude that the hyperbola does not intersect the circle.
  9 KB (1,646 words) - 22:56, 1 November 2024
• 2010 AMC 12A Problems/Problem 24
  ...egative. Continuing the pattern and accounting for doubled roots (which do not flip sign), we realize that there are <math>5</math> negative intervals fro ...count it as two disjoint intervals. Note that this will be important as to not undercounting disjoint intervals. )
  9 KB (1,434 words) - 16:54, 17 August 2022
• Python reference
  while word[0] not in vowels: ...erything from position 1, including position 1, all the way to position 3, not including position 3. This is called a slice.
  28 KB (4,762 words) - 20:20, 12 June 2023
• 2011 AMC 10B Problems/Problem 18
  ==Solution 2 (with trig, not recommended)== ==Solution 3(Easier Trig)==
  5 KB (842 words) - 19:57, 29 October 2024

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