كل شيء عن الدوال المثلثية sinx, cosx

1-4 معادلات مثلثية :

K ينتمي الى Z

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

cos(a) = cos(b)
ادن 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)


الرمز الذي لا يظهر في العهلاقات هو


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