สูตรตรีโกณมิติจากหนังสือคณิตศาสตร ปรนัย เล มที่ 31 โลก ...
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sin cos tan cosec sec cot
(-, ) (-, ) (-, ) - { | cos = 0} (-, ) - { | sin = 0} (-, ) - { | cos = 0} (-, ) - { | sin = 0}
[-1, 1] [-1, 1] (-, ) (-, -1] [1, ) (-, -1] [1, ) (-, )
arcsin arccos arctan arccot arcsec arccosec
[-1, 1] [-1, 1] (-, ) (-, ) (-, -1] [1, ) (-, -1] [1, )
[-
2
,
2
]
[0, ]
(-
2
,
2
)
(0, )
[0,
2
)
(
2
,
]
[-
2
,
0)
(0,
2
]
sin2 A + cos2 A = 1
1 + tan2 A = sec2 A
1 + cot2 A = cosec2 A
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
cot(A + B) =
cot A cot B -1 cot A + cot B
sin 2A = 2 sin A cos A
sin 2A
=
2 tan A 1 + tan2 A
cos 2A = cos2 A - sin2 A = 1 - 2 sin2 A = 2 cos2 A - 1
= sin2
A 2
1 - cos A 2
sin 3A = 3 sin A - 4 sin3 A
= cos2
A 2
1 + cos A 2
cos 3A = 4 cos3 A - 3 cos A
tan 2A =
2 tan A 1 - tan2 A
cos 2A
=
1 - tan2 A 1 + tan2 A
= tan2
A 2
1 - cos A 1 + cos A
sin(n + (-1) n ) = sin sin(n + ) = (-1) n sin
cos(2n ? ) = cos
tan(n - ) = -tan
cos(n + ) = (-1) n cos tan(n + ) = tan
sin A
+
sin B
=
2 sin
A+ B 2
cos
A-B 2
sin A
-
sin B
=
2 sin
A-B 2
cos
A+B 2
cos A
+
cos B
=
2 cos
A
+ B 2
cos
A-B 2
cos A
-
cos B
=
-2 sin
A+B 2
sin
A-B 2
2 sin A + B sin B - A
2
2
2 sin A cos B = sin(A + B) + sin(A - B) 2 cos A sin B = sin(A + B) - sin(A - B) 2 cos A cos B = cos(A + B) + cos(A - B) 2 sin A sin B = cos(A - B) - cos(A + B)
31
ABC a, b, c A, B, C
= ABC =
s(s - a)(s - b)(s - c)
s =
a+b+c 2
= 1 ab sin C = 1 ac sin B = 1 bc sin A
2
2
2
R = ABC r = ABC
= rs =
abc 4R
,
r
=
4R
sin
A 2
sin
B 2
sin
C 2
sin A a
=
sin B b
=
sin C c
=
2 abc
2R =
a sin A
=
b sin B
=
c sin C
R =
abc 4
c2 = a2 + b2 - 2ab cos C b2 = a2 + c2 - 2ac cos B a2 = b2 + c2 - 2bc cos A
cos A =
b2 + c2 - a2 2bc
cos B =
a2 + c2 - b2 2ac
cos C =
a2 + b2 - c2 2ab
sin
A 2
=
(s - b)(s - c) bc
sin
B 2
=
(s - a)(s - c) ac
sin
C 2
=
(s - a)(s - b) ab
cos A = s(s - a)
2
bc
cos B = s(s - b)
2
ac
cos C = s(s - c)
2
ab
tan
A 2
=
(s - b)(s - c) s(s - a)
tan
B 2
=
(s - a)(s - c) s(s - b)
tan
C 2
=
(s - a)(s - b) s(s - c)
ABC
( 5.)
1.
sin A
+ sin B
+ sin C
= 4 cos
A 2
cos
B 2
cos
C 2
2.
cos A
+
cos B
+ cos C
=
1 +
4 sin
A 2
sin
B 2
sin
C 2
3. tan A + tan B + tan C = tan A tan B tan C
sin, cos, tan
x1 ,
x2 , ... ,
x n
(0,
)
2
( 10.)
1.
sin x1 + sin x2 + ... + sin xn n
sin
x1
+
x2 + n
... +
xn
2.
n sin x1sin x2 ... sin xn
sin
x1
+
x2
+ ... + n
xn
3. cos x1 + cos x2 + ... + cos xn cos x1 + x2 + ... + xn
n
n
4.
n cos x1 cos x2 ... cos xn
cos x1 + x2 + ... + xn n
5.
tan x1 + tan x2 + ... + tan xn n
tan
x1
+
x2 + n
... +
xn
( x1 = x2 = ... = xn )
cos n sin
. 14.1 n
cos n =
- n + 2n-1 cosn 2n-3 cosn-2
n(n - 3) 2!
2n-5
cosn-4 -
n(n - 4)(n - 5) 3!
2n-7 cosn-6
+ - + ... n(n - 5)(n - 6)(n - 7) 4!
2n-9
cosn-8
n(n - 6)(n - 7)(n - 8)(n - 9) 5!
2n -11
cosn-10
. 13.4
sin
n
sin
2 n
sin
3 n
...
sin
(n
- 1) n
=
n 2n -1
n n > 1
. 14.17
sin
2n
sin
3 2n
sin
5 2n
...
sin
(n
- 4) 2n
sin
(n
- 2) 2n
=
1 2n -1
n
. 14.19
sin sin 3 sin 5 ... sin (n - 3) sin (n -1) =
2n 2n 2n
2n
2n
1 2n -1
n
................
................
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