كل شيء عن الدوال المثلثية sinx, cosx

1-1 علاقات أساسية :
tan(x) = sin(x)/cos(x)
sin²(x) cos²(x) = 1
sin²(x) = tan²(x) / (1 tan²(x))
cos²(x) = 1 / (1 tan²(x))
1-2 تحويلات شهيرة:
sin(2 x) = sin(x)
cos(2 x) = cos(x)
tan(2 x) = tan(x)

sin( -x) = - sin(x)
cos( -x) = cos(x)
tan( -x) = - tan(x)

sin( - x) = sin(x)
cos( - x) = - cos(x)
tan( - x) = - tan(x)

sin( x) = - sin(x)
cos( x) = - cos(x)
tan( x) = tan(x)

sin( /2 - x) = cos(x)
cos( /2 - x) = sin(x)
tan( /2 - x) = 1/tan(x)

sin( /2 x) = cos(x)
cos( /2 x) = - sin(x)
tan( /2 x) = -1/tan(x)

sin(3 /2 - x) = - cos(x)
cos(3 /2 - x) = - sin(x)
tan(3 /2 - x) = 1/tan(x)

sin(3 /2 x) = - cos(x)
cos(3 /2 x) = sin(x)
tan(3 /2 x) = -1/tan(x)

1-3 معادلات مثلثية :
k ينتمي الى Z

sin(a) = sin(b)
اذن a = b 2k
أو a = - b 2k

cos(a) = cos(b)
اذن a = b 2k
أو a = -b 2k

tan(a) = tan(b)
اذن a = b k
2- علاقات الجمع :
sin(a b) = sin(a)cos(b) sin(b)cos(a)
sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
cos(a b) = cos(a)cos(b) - sin(a)sin(b)
cos(a - b) = cos(a)cos(b) sin(a)sin(b)
tan(a b) = (tan(a) tan(b)) / (1 - tan(a)tan(b))
tan(a - b) = (tan(a) - tan(b)) / (1 tan(a)tan(b))

sin(p) sin(q) = 2sin((p q)/2)cos((p - q)/2)
sin(p) - sin(q) = 2sin((p - q)/2)cos((p q)/2)
cos(p) cos(q) = 2cos((p q)/2)cos((p - q)/2)
cos(p) - cos(q) = -2sin((p q)/2)sin((p - q)/2)
tan(p) tan(q) = sin(p q) / (cos(p)cos(q))
tan(p) - tan(q) = sin(p - q) / (cos(p)cos(q))

sin(a)sin(b) = (1/2)(cos(a - b) - cos(a b))
cos(a)cos(b) = (1/2)(cos(a b) cos(a - b))
sin(a)cos(b) = (1/2)(sin(a b) sin(a - b))

3- علاقات الضرب:
sin(2a) = 2sin(a)cos(a)
= 2tan(a) / (1 tan²(a))

cos(2a) = cos²a - sin²a
= 2cos²a - 1
= 1 - 2sin²a

tan(2a) = 2tan(a) / (1 - tan²(a))

sin²(a) = (1 - cos(2a)) / 2
cos²(a) = (1 cos(2a)) / 2
tan²(a) = (1 - cos(2a)) / (1 cos(2a))

tan(a) = sin(2a) / (1 cos(2a))
= (1 - cos(2a)) / sin(2a)

بوضع t = tan(a/2) :

sin(a) = 2t / (1 t²)
cos(a) = (1 - t²) / (1 t²)
tan(a) = 2t / (1 - t²)