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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. 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 ...

  4. 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.

  5. 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} .

  6. 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

  7. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π .

  8. Trigonometric polynomial - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_polynomial

    A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of ⁠ ⁠, or as a function on the unit circle.. Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm; [4] this is a special case of the Stone–Weierstrass theorem.

  9. cis (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Cis_(mathematics)

    cis is a mathematical notation defined by cis x = cos x + i sin x, [nb 1] where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form).