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  2. Differentiation of trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Differentiation_of...

    For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin( x ) and cos( x ) by means of the quotient rule applied to functions such ...

  3. Proofs of trigonometric identities - Wikipedia

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

    Illustration of the sum formula. Draw a horizontal line (the x -axis); mark an origin O. Draw a line from O at an angle α {\displaystyle \alpha } above the horizontal line and a second line at an angle β {\displaystyle \beta } above that; the angle between the second line and the x -axis is α + β {\displaystyle \alpha +\beta } .

  4. List of trigonometric identities - Wikipedia

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

    A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.

  5. Arctangent series - Wikipedia

    en.wikipedia.org/wiki/Arctangent_series

    The extremely slow convergence of the arctangent series for | | makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed ⁠ 1 4 π {\displaystyle {\tfrac {1}{4}}\pi } ⁠ as a sum of arctangents of smaller values, eventually resulting in a variety of ...

  6. Tangent half-angle formula - Wikipedia

    en.wikipedia.org/wiki/Tangent_half-angle_formula

    The sides of this rhombus have length 1. The angle between the horizontal line and the shown diagonal is ⁠ 1 / 2 ⁠ (a + b).This is a geometric way to prove the particular tangent half-angle formula that says tan ⁠ 1 / 2 ⁠ (a + b) = (sin a + sin b) / (cos a + cos b).

  7. Tangent half-angle substitution - Wikipedia

    en.wikipedia.org/wiki/Tangent_half-angle...

    The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = ⁡, sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: ⁡ = +, ⁡ = +, = +.

  8. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]

  9. Law of tangents - Wikipedia

    en.wikipedia.org/wiki/Law_of_tangents

    In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.