Half-Angle Formulas
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) = [画像:(-1)^(|_x/(2pi)_|)sqrt((1-cosx)/2)]
(1)
cos(1/2x) = [画像:(-1)^(|_(x+pi)/(2pi)_|)sqrt((1+cosx)/2)]
(2)
tan(1/2x) = [画像:(-1)^(|_x/pi_|)sqrt((1-cosx)/(1+cosx))]
(3)
= [画像:(sinx)/(1+cosx)]
(4)
= [画像:(1-cosx)/(sinx)]
(5)
= [画像:(tanxsinx)/(tanx+sinx)]
(6)
The corresponding hyperbolic function half-angle formulas are
sinh(1/2x) = [画像:sgn(x)sqrt((coshx-1)/2)]
(8)
cosh(1/2x) = [画像:sqrt((coshx+1)/2)]
(9)
tanh(1/2x) = [画像:(sinhx)/(coshx+1)]
(10)
= [画像:(coshx-1)/(sinhx).]
(11)
The Weierstrass substitution makes use of the half-angle formulas
cost = [画像:(1-tan^2(1/2t))/(1+tan^2(1/2t))]
(12)
sint = [画像:(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 classroomExplore with Wolfram|Alpha
WolframAlpha
More things to try:
Cite this as:
Weisstein, Eric W. "Half-Angle Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Half-AngleFormulas.html