العلاقات المثلثية




علاقات اساسية


tan(x) =
sin(x)/cos(x)
sin²(x) + cos²(x) = 1sin²(x) = tan²(x) / (1 + tan²(x))
cos²(x) = 1 / (1 +
tan²(x))
التحويلات الاساسية



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)


الزوايا الاساسية

x	sin(x)	cos(x)	tan(x)	cotan(x)
0	0	1	0	/
/6	1/2	(3)/2	(3)/3	(3)
/4	(2)/2	(2)/2	1	1
/3	(3)/2	1/2	(3)	(3)/3
/2	1	0	/	0
	0	-1	0	/

المعادلات المثلثية



R ينتمى الى k

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

cos(a) =
cos(b)
a = b + 2k
a = -b +
2k

tan(a) =
tan(b)
a = b + k


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))


علاقات اخرى



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)

t = tan(a/2) : بوضع

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


نظرية موافر



( cos(a) +
isin(a))ⁿ = cos(na) + isin(na)


نظرية اولر



cos θ = 1/2.(e^iθ +
e^-iθ)
sin θ = 1/(2i).(e^iθ -
e^-iθ)