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  • ...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>. ...\cos C</math>, so <math>\cos C = \frac{1}{8}</math>. Using the Pythagorean trig identity <math>\sin^2 + \cos^2 = 1</math>, <math>\sin^2 C = 1 - \frac{1}{64
    13 KB (2,116 words) - 23:24, 21 March 2024
  • ...math> and call its intersection with <math>DC</math> <math>K</math>. It is not hard to see that quadrilaterals <math>FGCK</math> and <math>JHKD</math> are .../math>. Now we have to find out what x is. For this, we break out a bit of trig. Let's look at <math>\triangle AFG</math>. By the law of sines:
    7 KB (1,145 words) - 17:59, 31 December 2023
  • ...tom-left segment. Then, it is easy to see that any point in the 5 segments not bordering the bottom-left segment will be distance at least <math>\dfrac{1} Last integral by trig substitution (long)
    12 KB (1,981 words) - 18:33, 3 September 2023
  • ==Solution 2 (No Trig)== Angle bisectors motivate trig bash.
    5 KB (906 words) - 17:43, 27 September 2023
  • ==Solution 1 (guys trig is fast)== ...f <math>m > 2</math>, then clearly <math>B</math> and <math>C</math> would not lie on the same side of <math>\ell</math>. Thus since <math>m > 0</math>, w
    31 KB (5,086 words) - 19:15, 20 December 2023
  • In this case, our base was one of the isosceles triangles (not the larger equilateral one). To calculate volume using the latter, note tha ==Solution 2 (No trig)==
    7 KB (1,074 words) - 01:49, 22 January 2024
  • ...{2}</math>. Multiply back the scalar and you get <math>\boxed{450}</math>. Not that hard, was it? ...\tan{\left(\alpha\right)} - d &= \sqrt{d^{2}-12} \end{align}</cmath> Using trig identities for the tangent, we find that <cmath>\begin{align*} \sqrt{3}\tan
    15 KB (2,560 words) - 01:44, 1 July 2023
  • ==Solution 1 (Uses Trig) == ...intuition, we can guess that the sidelength of the new triangle formed is not an integer, thus we pick <math>\boxed{\textbf{(E) } 37:1}</math>.
    13 KB (2,008 words) - 23:42, 17 July 2023
  • ==Solution 1 (Uses Trig) == ...intuition, we can guess that the sidelength of the new triangle formed is not an integer, thus we pick <math>\boxed{\textbf{(E) } 37}</math>.
    9 KB (1,416 words) - 14:30, 23 November 2023
  • ==Solution 2 (Trig Bash)== ==Solution 3 (Quicker Trig)==
    9 KB (1,539 words) - 15:47, 17 February 2024
  • ...preferably the largest value for each subset that works as they've special trig values and are the upper bound for the set with values of the <math>x</math
    3 KB (560 words) - 00:40, 1 June 2023
  • ==Solution 2 (Trig)== ...ath>. By cutting off the triangle of area <math>\frac{1}{2}</math> that is not part of the overlap, we get <math>\frac{\pi}{4} - \frac{1}{2} \approx \boxe
    3 KB (509 words) - 22:17, 25 February 2024
  • ==Solution 1 (No Trig)== ...ber that doubling the smallest angle of a 3-4-5 triangle gives the larger (not right) angle in a 7-24-25 triangle.
    14 KB (2,247 words) - 20:07, 12 January 2024
  • ...is that <math>xy</math> and <math>zx</math> have a similar structure, but not exactly conjugates, but instead once you take out the magnitudes of both, s ...i}{4}</math>. We need to convert the polar form to a standard form. Simple trig identities show <math>y=10+10i</math> and <math>z=3-3i</math>. More divisio
    11 KB (2,077 words) - 20:15, 12 January 2024
  • ...he intersection of circles <math>\omega_1</math> and <math>\omega_2</math> not equal to <math>A.</math> Then <math>AK=\tfrac mn,</math> where <math>m</mat == Solution 3 (Death By Trig Bash) ==
    12 KB (1,985 words) - 19:52, 28 January 2024
  • ...latively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> ...h>\triangle ABC</math>, or <math>\frac{3\sqrt{7}}{2}</math>. However, it's not too hard to see that <math>GB = HC = 1</math>, and therefore <math>[AGH] =
    35 KB (5,215 words) - 23:08, 29 October 2023
  • ...h>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> ...at they appear as in the diagram below. Note that <math>3HX = HY</math> is not insignificant; from here, we set <math>XH = HE = \frac{1}{2} EY = HL = 2</m
    16 KB (2,678 words) - 22:45, 27 November 2023
  • ...<math>\frac{s^2\sqrt{3}}{4}</math> (if you don't have this memorized it's not hard to derive). Comparing this formula to the area of <math>ABC</math>, we ...th> is equal to <math>1</math>, then we have <math>s=1</math>, but this is not possible since <math>P</math> is inside of the triangle. This means that <m
    16 KB (2,509 words) - 17:49, 8 February 2024
  • ...h>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 1 (No trig)==
    16 KB (2,517 words) - 20:22, 31 January 2024
  • ...ath>z=2\sin^2(\gamma)</math> (not necessarily this order, but here it does not matter due to symmetry), satisfying that <math>\alpha+\beta=180^{\circ}-\fr ==Solution 2 (pure algebraic trig, easy to follow)==
    15 KB (2,208 words) - 01:25, 1 February 2024

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