(ก
ก)
. ( )
ก
y
(0, 1)
%ก
&'
θ = 2nπ + π2 , n ∈ I
θ = π2 , 5π
, 9π , ...
2 2
S
A
(-1, 0)
(1, 0)
..., 7π, 5π, 3π, π = θ
%ก'(
T
θ = (2n + 1)π, n ∈ I
x
θ = 0, 2π, 4π, 6π, ...
C
%ก'
θ = 2nπ, n ∈ I
θ = 3π
, 7π , 11π , ...
2 2 2
(0,-1)
%ก
&)
θ = (2n + 1)π + π2 , n ∈ I
%ก'
θ = nπ, n ∈ I
%ก
&
θ = nπ + π2 , n ∈ I
กก (circular function)
1.
*+%ก',-. fnTri [%ก' ± θ ] = fnTri [θ]
2.
*+%ก
&,-.
fnTri [ %ก
& ± θ ] = Co-fnTri [θ]
:;&<=,= .>
2
*** 5
53o
71.5o
10
3
25
1
37o
24
18.5o
4
74o
3
7
63.5o
5
16o
50
82o
1
1
8o
26.5o
7
2
2 2
3 −1
sin 15
=
cos 15
=
tan 15
=
3 −1
cot 15
=
3 +1
2 2
3 +1
2 2
3 -1
15?
3 +1
3 +1
3 −1
sin 22.5
=
1
2
2− 2
tan 22.5
=
2 −1
cos 22.5
=
1
2
2+ 2
cot 22.5
=
2 +1
4
18?
10 + 2 5
5-1
4
5+1
54?
10 - 2 5
3
=
2− 3
=
2+ 3
FORMULAS
+,ก (Compound Angle)
1. sin(A + B) = sin A cos B + cos A sin B ,
sin(A − B)
=
sin A cos B − cos A sin B
2.
cos(A + B)
=
cos A cos B − sin A sin B
,
cos(A − B)
=
cos A cos B + sin A sin B
3.
tan(A + B)
=
tan A+tan B
1−tan A tan B
,
tan(A − B)
=
tan A−tan B
1+tan A tan B
4.
cot(A + B)
=
cot A cot B−1
cot B+cot A
,
cot(A − B)
=
cot A cot B+1
cot B−cot A
2 5 (DOUBLE ANGLE)
5. sin 2A = 2 sin A cos A
6.
cos 2A
=
2 tan A
1+tan 2 A
=
=
=
=
cos 2 A − sin 2 A
2 cos 2 A − 1
1 − 2 sin 2 A
1−tan 2 A
1+tan 2 A
7.
tan 2A
=
2 tan A
1−tan 2 A
8.
cot 2A
=
cot 2 A−1
2 cot A
4
3 5 (TRIPLE ANGLE)
9. sin 3A = 3 sin A − 4 sin 3 A
10.
cos 3A
=
4 cos 3 A − 3 cos A
11.
tan 3A
=
3 tan A−tan 3 A
1−3 tan 2 A
12.
cot 3A
=
3 cot A−cot 3 A
1−3 cot 2 A
Cก , CE5 , F,C
G (SUM , DIFFERENCE , AND PRODUCT)
(A−B)
13. sin A + sin B = 2 sin (A+B)
cos 2
2
14.
sin A − sin B
=
2 cos
(A+B)
(A−B)
sin 2
2
15.
cos A + cos B
=
2 cos
(A+B)
(A−B)
cos 2
2
* 16.
cos A − cos B
=
−2 sin
17.
2 sin A cos B
=
sin(A + B) + sin(A − B)
18.
2 cos A sin B
=
sin(A + B) − sin(A − B)
19.
2 cos A cos B
=
cos(A + B) + cos(A − B)
2 sin A sin B
=
cos(A − B) − cos(A + B)
* 20.
(A+B)
(A−B)
sin
2
2
5
!ก
ก
y = arc sin x ,
−
y
−1 ≤ x ≤ 1 , −π2 ≤ y ≤ π2
π
2
** K+
sin −1 x + sin −1 (−x) = 0
sin −1 (−x) =
0
(1, 0)
− sin −1 x
x
− π2
y = arc cos x ,
** K+
y
−1 ≤ x ≤ 1 , 0 ≤ y ≤ π
cos −1 x + cos −1 (−x) = π
π
cos −1 (−x) = π − cos −1 x
y = arc tan x ,
x ∈ R , −π2 < y < π2
0
(1, 0)
x
y
π
2
** K+
tan −1 x + tan −1 (−x) = 0
tan −1 (−x) = − tan −1 x
0
(1, 0)
− π2
6
x
กLกK
1. sin (sin −1 x)
sin −1 (sin x)
2.
= x , x ∈ [−1, 1]
= x , x ∈ [ −π2 , π2 ]
cos (cos −1 x) = x , x ∈ [−1 , 1]
cos −1 (cos x) = x , x ∈ [0, π]
3.
tan (tan −1 x) = x , x ∈ R
tan −1 (tan x)
= x , x ∈ ( −π2 , π2 )
กLก arc
x+y
1. arctan x + arctan y = arctan 1−xy
;&<
xy < 1
2.
arctan x + arctan y = arctan 1−xy + π
x+y
;&<
xy > 1 , x > 0
%)
y>0
3.
arctan x + arctan y = arctan 1−xy − π
x+y
;&<
xy > 1 , x < 0
%)
y<0
4.
arctan x − arctan y = arctan 1+xy
x−y
;&<
xy > 0
กLก+M arc
5
0
3
θ = arcsin 35 = arccos 45 = arctan 34 = arccot 43 = arcsec 54 = arccsc 53
4
KENONEN
1. arcsin x + arccosx
= π2
4.
arcsin x = arccsc 1x
2.
arctan x + arccot x = π2
5.
arccos x = arcsec 1x
3.
arc sec x + arccsc x = π2
6.
arctan x = arccot 1x , x > 0
7
ก"
#ก$!" ,,F,
K
กL sine (LAW of sine)
C
a = b = c
sin A
sin B sin C
a
b
A
+,กE
sin A = sin B = sin C
a
c
b
B
c
sin A : sin B : sin C = a : b : c
กL cosine (LAW of cosine)
a 2 = b 2 + c 2 − 2bc cos A
C
A
b 2 = c 2 + a 2 − 2ca cos B
a
b
B
c
c 2 = a 2 + b 2 − 2ab cos C
กLR+
(LAW of Projection)
C
a
b
A
c
B
A
b
c
B
b cos A + a cos B = c
B
a
c cos B + b cos C = a
8
c
a
C
C
b
a cos C + c cos A = b
A
กก
1. ก
f(x) =
sin x
1 − cos 2 x
+
cos x
1 − sin 2 x
+
tan x
sec 2 x − 1
+
cot x
csc 2 x − 1
x !
ก"#ก $ !ก%&"'$(
1. −4
2. −2
3. 2
4. 4
2. ,& θ !$-."!$ 1 /01 3 cos θ − 4 sin θ
3 sin θ + 4 cos θ $-.""'$(
1. (1, 2]
2. (2, 3]
3. (3, 4]
9
= 2
/0&
4. (4, 5]
3. ก6"6& A, B, C /01 D "
[0, π]
#-!$
sin A + 7 sin B = 4(sin C + 2 sin D)
cos A + 7 cos B = 4(cos C + 2 cos D)
cos (A − D)
cos (B − C)
!ก%!"
4. "6& ABC .60$- #-!$
1. 16
65
cos C
sin A = 35
/01
!ก%&"'$( (PAT 1 $.. 54)
2. − 16
3. 48
65
65
10
5
cos B = 13
4.
− 33
65
5. ,& cos 0 1cos 1
+ cos 1 1cos 2 + cos 2 1cos 3 + ..... + cos 44 1cos 45 = sec A
/01 0 < A < 180
/0& A $!ก%!"
6. ก6"6&
0 < θ < 45
/01"6&
A = (sin θ) tan θ , B = (sin θ) cot θ , C = (cot θ) sin θ , D = (cot θ) cos θ
&"'$( ก (PAT 1 $.. 55)
1. A < B < C < D
3. A < C < D < B
2. B < A < C < D
4. C < D < B < A
11
7. Given that
(1 + tan 1 )(1 + tan 2 )...(1 + tan 45 ) = 2 n ,
find n.
8. log 2 (1 + tan 1 ) + log 2 (1 + tan 2 ) + ..... + log 2 (1 + tan 44 ) !ก%!"
(PAT 1 $.. 54)
12
9. ,&
x
1 − cot 20 = 1 − cot
25
/0& x $!" (PAT 1 .. 52)
10. ก6"6& (1 − cot 1 )(1 − cot 2 )(1 − cot 3 ).....(1 − cot 44 )
x 2 6&- 10 60NO!ก%!"
13
= x
11. ,&
1.
cos θ − sin θ =
5
3
/0&
4
13
2.
9
13
sin 2θ
$ก%&" (PAT 1 $.. 52)
3.
4
9
4.
13
9
12. ,& sin 15 /01 cos 15 %ก x 2 + ax + b = 0
/0& a 4 − b !ก%&"'$( (PAT 1 ก.. 53)
1. −1
2. 1
3. 2
4. 1 + 3 2
Suppose the roots of the quadratic equation x 2 + ax + b = 0 are sin 15 and cos 15
What is the value of a 4 − b ?
14
13. ก6"6&.60$- ABC $
A /01
B /60
,& cos 2A + 3 cos 2B = −2 /01 cos A − 2 cos B = 0
/0& cos C $!ก%&"'$( (% .. 55)
1. 15 ( 3 − 2 ) 2. 15 ( 3 + 2 ) 3. 15 (2 3 − 2 ) 4.
5.
1
(2
5
2 − 3)
14. 6
lim
(cot 3 x − 1) cosec 2 x
2
x → π 1 + cos 2x − 2 sin x
(PAT 1 $.. 55)
4
15
1
(
5
2 +2 3 )
15. ก6"6& a /010&ก%ก
5(sin a + cos a) + 2 sin a cos a = 0.04
16. 125(sin 3 a + cos 3 a) + 75 sin a cos a
!ก%!" (PAT 1 .. 53)
12 cot 9 (4 cos 2 9 − 3)(4 cos 2 27 − 3)
16
!ก%!"
17. 6
18. 2 sin 2 60 (tan 5 + tan 85 ) − 12 sin 70
cos 36 − cos 72
sin 36 tan 18 + cos 36
(PAT 1 $.. 55)
!ก%!" (PAT 1 $.. 53)
17
19. tan 20 + 4 sin 20
sin 20 sin 40 sin 80
!ก%!" (PAT 1 .. 54)
20. Suppose that a and b is a non-zero real number for which
sin x + sin y = a and cos x + cos y = b, What is the value of sin (x + y) ?
2ab
a2 − b2
a2 + b2
1. a 22ab
2.
3.
4.
5.
2ab
2ab
− b2
a2 + b2
18
a2 − b2
a2 + b2
21. ก6"6&
 cosec 10
A = 
 sec 10
 cos 2 70 sin 40 
3 
, B = 

1 
0
cos 2 50 

 cos 2 20

0
C = 

2
 sin 80 cos 10 
det [A(B + C)]
!ก%!" (PAT 1 $.. 54)
19
/01
22. c0$-0d
1. cot 1
2.
23. ,&
2 sin 2 , 4 sin 4 , 6 sin 6 , ....., 180 sin 180
csc 1
3.
3 tan A = tan (A + B)
/0&
cot 2
sin (2A + B)
sin B
20
4.
csc 2
$!"
$!ก%&"
5. tan 1
24. "6& x /01 y ef arcsin(x + y) + arccos(x − y) = 3π2
( arcsin y + arccos x $-.""'$(
1.
[− π , 0)
2
2.
[0, π ]
2
3.
( π , π]
2
4.
(π, 3π ]
2
25. ,& arcsin 5x + arcsin x = π2 /0& tan(arcsin x) !ก%&"'$(
(PAT 1 ก.. 52)
1. 15
2. 13
3. 13
4. 12
21
26. ,& (arctan 1 + arctan 2 + arctan 3)
/0& K $!ก%&"'$(
1. 2
2. 3
27. = K(arctan 1 + arctan 12 + arctan 13 )
tan[arc cot 1 − arc cot 1 + arc tan 7 ]
5
3
9
5
12
sin  arc sin + arc sin 

13
13 
3. 4
4. 5
!ก%!" (PAT 1 .. 53)
22
28. cot (arc cot 7 + arc cot 13 + arc cot 21 + arc cot 31) !ก%&"'$(
(PAT 1 $.. 54)
1. 114
2. 134
3. 92
4. 252
29. tan (arc cot 3 + arc cot 7 + arc cot 13 + arc cot 21 + arc cot 31 + arc cot 43)
!ก%!"
23
30.  10

cot  Σ arctan 12 
2n 
n = 1
!ก%!"
31. ก6"6& C = arcsin 35 + arc cot 53 − arctan 198
,& A e%ก arc cot 2x1 + arc cot 3x1 = C
/0&h0.dก"e A !ก%&"'$( (PAT 1 .. 54)
1. − 14
2. 14
3. − 16
4. 16
24
32. "6& A e%ก
arccos (x) = arccos(x 3 ) + arccos( 1 − x 2 )
/01"6& B e%ก arccos (x) = arcsin (x) + arcsin (1 − x)
กe P(A − B) !ก%!" P(S) /!iee S
(PAT 1 $.. 55)
25
33. ,& A e%ก
2
3 arcsin  2x 2  − 4 arccos  1 − x 2  + 2 arctan  2x 2  = π3
1+x
1+x
1−x
/0&กe A !ก%!"
34. ,&
0 < θ < π4
/01
tan 2 θ + 4 tan θ − 1 = 0
26
/0&
tan [arccos (sin 4θ)]
$!"
35.
sec 2 (2 tan −1 2 )
$!ก%!" (% .. 55)
36. 60$- .6f $& - 2x + 3, x 2 + 3x + 3, x 2 + 2x 0% /0&
!$" 6j!$ 60$- .$( $ ก$ N
27
37. "6& ABC .60$-#-$ a, b /01 c -&&
A B /01
C 0% ,&
C !ก% 60 b = 5 /01 a − c = 2
/0&-&%.60$- ABC !ก%&"'$( (PAT 1 $.. 55)
1. 25
2. 29
3. 37
4. 45
38. ก6"6& A, B /01 C .60$- #-!$
(sin A − sin B + sin C)(sin A + sin B + sin C) = 3 sin A sin C
6
3 sec 2 B + 3 cosec 2 B
(PAT 1 .. 54)
28
39. .60$- ABC $-&&
A, B /01 C a 6-, b 6/01 c 6- 0%
,& a 2 + b 2 = 49c 2 /0& cot Acot+ Ccot B !ก%!"
40. ก6"6& ABC .60$-"k $-&
A, B /01 C a, b /01 c 6-0%
,& a 2 + b 2 = 31c 2 /0& 3 tan C(cot A + cot B) !ก%!" (PAT 1 $.. 54)
29
41. .60$- ABC $& a, b, c &&
A, B, C ef$-
3, 2.5, 1 0% b cos C + c cos B !ก%!"
42. ก660$- ABC $ A /01 B /60 ,&
sin A = 3 , cos B = 5
& a - 13 6- & c -!ก%!" (TEXT BOOK)
30
5
13
43. "6& ABC .60$-#-$ a, b /01 c -&&
A B /01
C 0%
,& c = 5 /0& (a − b) 2 cos 2 C2 + (a + b) 2 sin 2 C2 !ก%!"
44. .60$- ABC $-&&
A, B /01 C a 6-, b 6/01 c 6- 0% ,&
A, B /01 C 0%0d /01 b : c = 3 : 2
/0& tan A + cot A !ก%!"
31
45. ".60$- ABC "k ,& a, b /01 c -&&
A B
/01
C 0% /0& c cos Bcos+ Ab cos C + a cos Ccos+ cBcos A + b cos Acos+Ca cos B
!ก%&"'$(
2 b2 + c2
1. a +2abc
2.
(a + b + c) 2
abc
3.
(a + b + c) 2
2abc
46. ก6"6& ABC 60$- #-!$ B = π2 + A
/01 BC = x , AC = y ef0&' x y = y x /01
/0& cos 2A $!"
32
4.
a2 + b2 + c2
abc
y = x2
99. i$ & --.! !N1กfก606f 6o-fก
- 45 ก
$(
i$ & '!N"&1-1 100 16o-fก (!$ /6) 30 .fก!ก%&"'$( (Ent)
1. 100
2. 50 2
3. 50 3
4. 1003
W
A
N
B
45r
30r
S
C
E
100 D
100.i$ p- -.! !N1กq%/66f /016o-q
60 N / '!!N61-1! x i$ p,6o
-q
-i$- 45 N ,&i$p/01q. 1.60 /01 37.60 0% /0& x 2 -.""'$(
1. (0, 250]
2. (250, 500]
3. (500, 750]
4. (750, 1000]
33