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  • 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) - 21:55, 20 January 2024
  • ==Solution 2 (Trig)== ==Solution 4 (Trig)==
    11 KB (1,741 words) - 22:40, 23 November 2023
  • ...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
    19 KB (3,221 words) - 01:05, 7 February 2023
  • == 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) - 23:44, 12 October 2023
  • ...>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)==
    3 KB (534 words) - 03:22, 23 January 2023
  • ...th> where <math>m</math> and <math>n</math> are positive integers and n is not divisible by the square of any prime. Find <math>m + n.</math> ==Solution 4 (No Trig)==
    9 KB (1,461 words) - 15:09, 18 August 2023
  • ==Solution 2 (no trig)== ==Solution 3 (trig)==
    6 KB (854 words) - 20:25, 24 July 2022
  • ...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) - 02:36, 19 March 2022
  • 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
    12 KB (2,015 words) - 20:54, 9 October 2022
  • ...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) - 22:48, 7 March 2024
  • ...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) - 07:09, 22 December 2022
  • ...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 (1,622 words) - 20:53, 11 September 2023
  • ...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) - 17:54, 17 August 2022
  • 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) - 21:20, 12 June 2023
  • ==Solution 2 (with trig, not recommended)== ==Solution 3(Easier Trig)==
    5 KB (782 words) - 23:04, 31 August 2023
  • == Solution 2 (trig) == ...e time in seconds. Since <math>x=0</math> is at the crest of the graph and not at the midline, we will use a cosine graph. Therefore, we will use the form
    3 KB (559 words) - 02:44, 8 February 2024
  • ...d \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math> .../math> to get that <math>BD=\frac{c\cos{\beta}}{2}</math>. We can also use trig manipulation on <math>BCE</math> to get that <math>CE=a\tan{\beta}</math>.
    2 KB (303 words) - 20:28, 2 October 2023
  • ...latively prime positive integers, and <math>p</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> ...ector Theorem or the Law of Cosines, but it uses the Law of Sines and more trig)
    9 KB (1,523 words) - 12:23, 7 September 2022
  • ...</math>, where <math>a,b,c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> == Solution 2 (Trig) ==
    5 KB (831 words) - 17:55, 21 July 2018
  • ...h>, where <math>a</math> and <math>d</math> are relatively prime, and c is not divisible by the square of any prime. Find <math>a+b+c+d</math>. ...ath>H</math>, and note that <math>QP = 1 + \sqrt{3}</math> after using the trig functions for <math>75</math> degrees.
    7 KB (1,068 words) - 12:11, 10 August 2021
  • ...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
  • == Video Solution by OmegaLearn (Extending Lines, Angle Chasing, Trig Area) == ...er_spindle , and can be used to demonstrate that <math>3</math> colors are not sufficient to color all of the points in the plane such that points that ar
    4 KB (537 words) - 13:49, 25 February 2024
  • ...m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math> ==Solution 2 (Trig) ==
    8 KB (1,279 words) - 12:07, 28 October 2023
  • (Note that the cevians do not necessarily lie within the triangle, although they do in this diagram.) The proof using [[Routh's Theorem]] is extremely trivial, so we will not include it.
    5 KB (934 words) - 13:06, 20 February 2024
  • ...\sin^4 \frac{2 \pi}{7} + 7 \cdot 2^4 \sin^4 \frac{3 \pi}{7}</math>. Using trig we get ...as]], and the desired identity follows. See [[Roots of unity]] if you have not seen this technique.
    21 KB (3,265 words) - 17:06, 15 November 2023
  • ...th>, <math>n</math>, and <math>p</math> are integers and <math>n</math> is not divisible by the square of any prime. ...circ)</math> is by memory, but we can cleverly calculate it using a common trig identity:
    20 KB (2,980 words) - 18:17, 2 January 2024
  • ...th that is <math>120</math> meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could ...right triangles (including <math>\triangle{RSC}</math>) will have the same trig ratios. By proportion, the hypotenuse <math>AP</math> is <math>\frac{x}{100
    7 KB (1,074 words) - 21:22, 20 November 2023