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  2. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    The line segment ¯ has length ⁡ and sum of the lengths of ¯ and ¯ equals the length of ¯, which is 1. Therefore, cos ⁡ 2 θ + 2 sin 2 ⁡ θ = 1 {\displaystyle \cos 2\theta +2\sin ^{2}\theta =1} .

  3. Pythagorean trigonometric identity - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_trigonometric...

    A calculation confirms that z(0) = 1, and z is a constant so z = 1 for all x, so the Pythagorean identity is established. A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine.

  4. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [ 1 ] In the table below, the label "Undefined" represents a ratio 1 : 0. {\displaystyle 1:0.}

  5. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have ⁡ < <. For negative values of θ we have, by the symmetry of the sine function

  6. File:Sine cosine one period.svg - Wikipedia

    en.wikipedia.org/wiki/File:Sine_cosine_one...

    English: SINE and COSINE-Graph of the sine- and cosine-functions sin(x) and cos(x). One period from 0 to 2π is drawn. x- and y-axis have the same units. All labels are embedded in "Computer Modern" font. The x-scale is in appropriate units of pi.

  7. Sine and cosine - Wikipedia

    en.wikipedia.org/wiki/Sine_and_cosine

    The fixed point iteration x n+1 = cos(x n) with initial value x 0 = −1 converges to the Dottie number. Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin ⁡ ( 0 ) = 0 {\displaystyle \sin(0)=0} .

  8. College Football Playoff quarterfinal schedule, matchups ...

    www.aol.com/college-football-playoff-odds-match...

    We have a new favorite to win the College Football Playoff after the first round.. No. 5 seed Texas is +300 to win it all at BetMGM after its 38-24 win over No. 12 Clemson on Saturday. The ...

  9. Table of the largest known graphs of a given diameter and ...

    en.wikipedia.org/wiki/Table_of_the_largest_known...

    Below is the table of the vertex numbers for the best-known graphs (as of July 2022) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10.