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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...

    The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.

  4. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    The expression cos x + i sin x is sometimes abbreviated to cis x. The formula is important because it connects complex numbers and trigonometry . By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x ...

  5. Proofs of trigonometric identities - Wikipedia

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

    This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] 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 ...

  6. Trigonometric functions of matrices - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions_of...

    The analog of the Pythagorean trigonometric identity holds: [2] ⁡ + ⁡ = If X is a diagonal matrix, sin X and cos X are also diagonal matrices with (sin X) nn = sin(X nn) and (cos X) nn = cos(X nn), that is, they can be calculated by simply taking the sines or cosines of the matrices's diagonal components.

  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. Could Matt Gaetz decide to keep his House seat? It's ... - AOL

    www.aol.com/could-matt-gaetz-decide-keep...

    Gaetz resigned from the current Congress, but he was re-elected in November to represent his Florida district. That raises the question: Could he return in January?

  9. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.