# Mathc initiation/a78a

Comparons les équations : :
| Trigonométriques | Trigonométriques hyperboliques |
| ***************** * cos(x)**2 + sin(x)**2 = 1 * sec(x)**2 - tan(x)**2 = 1 * csc(x)**2 - cot(x)**2 = 1 ***************** sin(x+y) = cos(x) sin(y) + sin(x) cos(y) cos(x+y) = cos(x) cos(y) - sin(x) sin(y) sin(x-y) = sin(x) cos(y) - cos(x) sin(y) cos(x-y) = cos(x) cos(y) + sin(x) sin(y) tan(x+y) = (tan(x)+tan(y)) / (1-tan(x)*tan(y)) tan(x-y) = (tan(x)-tan(y)) / (1+tan(x)*tan(y)) ***************** cos(2x) = cos(x)**2 - sin(x)**2 = 1 - 2sin(x)**2 = 2cos(x)**2 - 1 cos(3x) = 4cos(x)**3 - 3cos(x) cos(4x) = 8cos(x)**4 - 8cos(x)**2 + 1 sin(2x) = 2cos(x) sin(x) sin(3x) = 3sin(x) - 4sin(x)**3 sin(4x) = 4sin(x) cos(x) - 8sin(x)**3 cos(x) tan(2x) = (2*tan(x)) / (1- tan(x)**2) tan(3x) = (3tan(x)-tan(x)**3) / (1-3tan(x)**2) ***************** cos(x)**2 = 1/2 + 1/2 cos(2x) cos(x)**3 = 3/4 cos(x) + 1/4 cos(3x) cos(x)**4 = 3/8 + 1/2 cos(2x) + 1/8 cos(4x) sin(x)**2 = 1/2 - 1/2 cos(2x) sin(x)**3 = 3/4 sin(x) - 1/4 sin(3x) sin(x)**4 = 3/8 - 1/2 cos(2x) + 1/8 cos(4x) ***************** sin(x/2) = sqrt( (1-cos(x))/2) cos(x/2) = sqrt( (1+cos(x))/2) tan(x/2) = sqrt( (1-cos(x)) / (1+cos(x)) ) ***************** sin(x)sin(y) = 1/2 [cos(x-y) - cos(x+y)] cos(x)cos(y) = 1/2 [cos(x-y) + cos(x+y)] sin(x)cos(y) = 1/2 [sin(x-y) + sin(x+y)] ***************** sin(x) + sin(y) = 2 sin((x+y)/2) cos((x-y)/2) sin(x) - sin(y) = 2 sin((x-y)/2) cos((x+y)/2) cos(x) + cos(y) = 2 cos((x+y)/2) cos((x-y)/2) cos(x) - cos(y) = -2 sin((x+y)/2) sin((x-y)/2) = 2 sin((x+y)/2) sin((y-x)/2) ***************** sin(acos(x)) = sqrt(1-x**2) cos(asin(x)) = sqrt(1-x**2) sec(atan(x)) = sqrt(1+x**2) csc(acot(x)) = sqrt(1+x**2) tan(asec(x)) = sqrt(x**2-1) cot(acsc(x)) = sqrt(x**2-1) ***************** | ***************** * cosh(x)**2 - sinh(x)**2 = 1 * sech(x)**2 + tanh(x)**2 = 1 * coth(x)**2 - csch(x)**2 = 1 ***************** sinh(x+y) = cosh(x) sinh(y) + sinh(x) cosh(y) cosh(x+y) = cosh(x) cosh(y) + sinh(x) sinh(y) sinh(x-y) = sinh(x) cosh(y) - cosh(x) sinh(y) cosh(x-y) = cosh(x) cosh(y) - sinh(x) sinh(y) tanh(x+y) = [tanh(x)+tanh(y)]/[1+tanh(x)*tanh(y)] tanh(x-y) = [tanh(x)-tanh(y)]/[1-tanh(x)*tanh(y)] ***************** cosh(2x) = cosh(x)**2 + sinh(x)**2 = 1 + 2*sinh(x)**2 = 2*cosh(x)**2 - 1 cosh(3x) = 4*cosh(x)**3 - 3*cosh(x) cosh(4x) = 8 cosh(x)**4 - 8 cosh(x)**2 + 1 sinh(2x) = 2 cosh(x) sinh(x) sinh(3x) = 3 sinh(x) + 4*sinh(x)**3 sinh(4x) = 4 cosh(x) sinh(x) + 8 sinh(x)**3 cosh(x) tanh(2x) = (2 tanh(x))/(1 + tanh(x)**2) tanh(3x) = (3tanh(x)+tanh(x)**3) /(1+3tan(x)**2) ***************** cosh(x)**2 = 1/2 + 1/2 cosh(2x) cosh(x)**3 = 3/4 cosh(x) + 1/4 cosh(3x) cosh(x)**4 = 3/8 + 1/2 cosh(2x) + 1/8 cosh(4x) sinh(x)**2 = 1/2 cosh(2x) - 1/2 sinh(x)**3 = 1/4 sinh(3x) - 3/4 sinh(x) sinh(x)**4 = 3/8 - 1/2 cosh(2x) + 1/8 cosh(4x) ***************** ***************** sinh(x) sinh(y) = 1/2 [cosh(x+y) - cosh(x-y)] cosh(x) cosh(y) = 1/2 [cosh(x-y) + cosh(x+y)] sinh(x) cosh(y) = 1/2 [sinh(x-y) + sinh(x+y)] ***************** sinh(x) + sinh(y) = 2 sinh((x+y)/2) cosh((x-y)/2) sinh(x) - sinh(y) = 2 sinh((x-y)/2) cosh((x+y)/2) cosh(x) + cosh(y) = 2 cosh((x+y)/2) cosh((x-y)/2) cosh(x) - cosh(y) = 2 sinh((x+y)/2) sinh((x-y)/2) ***************** sinh(acosh(x)) = sqrt(x**2 - 1) cosh(asinh(x)) = sqrt(x**2 + 1) tanh(asech(x)) = sqrt(1 - x**2) coth(acsch(x)) = sqrt(1 + x**2) ***************** |