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Half-Angle Formulas

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Half-angle formulas and formulas expressing trigonometric functions of an angle x/2 in terms of functions of an angle x. For real x,

sin(1/2x)=(-1)^(|_x/(2pi)_|)sqrt((1-cosx)/2)
(1)
cos(1/2x)=(-1)^(|_(x+pi)/(2pi)_|)sqrt((1+cosx)/2)
(2)
tan(1/2x)=(-1)^(|_x/pi_|)sqrt((1-cosx)/(1+cosx))
(3)
=(sinx)/(1+cosx)
(4)
=(1-cosx)/(sinx)
(5)
=(tanxsinx)/(tanx+sinx)
(6)
=((-1)^(|_(x+pi/2)/pi_|)sqrt(1+tan^2x)-1)/(tanx).
(7)

The corresponding hyperbolic function half-angle formulas are

sinh(1/2x)=sgn(x)sqrt((coshx-1)/2)
(8)
cosh(1/2x)=sqrt((coshx+1)/2)
(9)
tanh(1/2x)=(sinhx)/(coshx+1)
(10)
=(coshx-1)/(sinhx).
(11)

The Weierstrass substitution makes use of the half-angle formulas

cost=(1-tan^2(1/2t))/(1+tan^2(1/2t))
(12)
sint=(2tan(1/2t))/(1+tan^2(1/2t)).
(13)

See also

Double-Angle Formulas, Hyperbolic Functions, Multiple-Angle Formulas, Prosthaphaeresis Formulas, Trigonometric Addition Formulas, Trigonometric Functions, Trigonometry, Weierstrass Substitution Explore this topic in the MathWorld classroom

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Cite this as:

Weisstein, Eric W. "Half-Angle Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Half-AngleFormulas.html

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