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  2. List of integrals of trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/List_of_integrals_of...

    The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals.

  3. List of trigonometric identities - Wikipedia

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

    An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

  4. Tangent half-angle substitution - Wikipedia

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

    The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. [2] Leonhard Euler used it to evaluate the integral ∫ d x / ( a + b cos ⁡ x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook , [ 3 ] and Adrien-Marie Legendre described ...

  5. Trigonometric substitution - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_substitution

    Lists of integrals; Integral transform; Leibniz integral rule; Definitions; Antiderivative; Integral Riemann integral; Lebesgue integration; Contour integration; Integral of inverse functions; Integration by; Parts; Discs; Cylindrical shells; Substitution (trigonometric, tangent half-angle, Euler) Euler's formula; Partial fractions (Heaviside's ...

  6. Integral of the secant function - Wikipedia

    en.wikipedia.org/wiki/Integral_of_the_secant...

    Twice the area of the purple triangle is the stereographic projection s = tan ⁠ 1 / 2 ⁠ ϕ = tanh ⁠ 1 / 2 ⁠ ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos ϕ, sin ϕ). The purple point has coordinates (0, s). The integral of the hyperbolic secant function defines the Gudermannian function:

  7. Proofs of trigonometric identities - Wikipedia

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

    The sign of the square root needs to be chosen properly—note that if 2 π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ. For the tan function, the equation is:

  8. Tangent half-angle formula - Wikipedia

    en.wikipedia.org/wiki/Tangent_half-angle_formula

    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). The formulae sin ⁠ 1 / 2 ⁠ (a + b) and cos ⁠ 1 / 2 ⁠ (a + b) are the ratios of the actual distances to ...

  9. Trigonometric integral - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_integral

    Integral sine in the complex plane, plotted with a variant of domain coloring. Integral cosine in the complex plane. Note the branch cut along the negative real axis. In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.