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  2. List of trigonometric identities - Wikipedia

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

    The angle difference identities for ⁡ and ⁡ can be derived from the angle sum versions by substituting for and using the facts that ⁡ = ⁡ and ⁡ = ⁡ (). They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

  3. Proofs of trigonometric identities - Wikipedia

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

    The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:

  4. Prosthaphaeresis - Wikipedia

    en.wikipedia.org/wiki/Prosthaphaeresis

    Sum and difference: Find the sum and difference of the two angles. Average the cosines : Find the cosines of the sum and difference angles using a cosine table and average them, giving (according to the second formula above) the product cos ⁡ α cos ⁡ β {\displaystyle \cos \alpha \cos \beta } .

  5. Trigonometry - Wikipedia

    en.wikipedia.org/wiki/Trigonometry

    Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. [83] Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities, [84] relate both the sides and angles of a given triangle.

  6. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes sin ⁡ ( 30 ∘ ) = 1 / 2 {\displaystyle \sin(30^{\circ ...

  7. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

  8. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    See angle sum and difference identities. We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0x) + i sin(0x) = 1 + 0i = 1. Finally, for the negative integer cases, we consider an exponent of −n for natural n.

  9. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting -tuple—over the harmonic series, advancing the window by positions to select the next -tuple, and offsetting each element of each tuple by relative to the window's ...