Trigonometry 1 (key)

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1
1. sin 214π cos( 34π π) + cos 214π sin( 34π π)
! ก
ก !!" A = 214π , B = 34π π
#$%%
& sin A ⋅ cos B + cos A ⋅ sin B '()!#ก% sin (A + B)
*+ sin 214π cos ( 34π π) + cos 214π sin ( 34π π)
= sin ( 214π + 34π π)
= sin (6π π) = sin 5π = 0
2.
cot ( π4 2π) ⋅ sec ( 23π π2 ) ⋅ tan (arctan 1)
! -)ก cot ( π4 2π) ⋅ sec ( 23π π2 ) ⋅ tan (arctan 1)
= cot (2π π4 ) ⋅ sec π6
= cot π4 ⋅sec π6
= 1⋅ 2
= 2
3.
'
#)ก% 18.84 (ก23 π ≈ 3.14)
! ก%
sin 18.84
-)ก sin 18.84 = sin (6 ⋅ 3.14)
= sin 6π
= 0
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ก.
4.
4
56ก7 f(x) = 3 sin x2 + 4 cos x2 37 0 < x ≤ π
d
du
(ก23 dud (cos u) = sin u ⋅ du
dx $ du (sin u) = cos u ⋅ dx )
! 68#37!-)4*9
56ก7!7$ก
$ก 4*9%() f′(x) = 23 cos x2 2 sin x2
3
f′(x) = 0
23 cos x2 2 sin x2 = 0
3 cos x2 4 sin x2 = 0
tan x2 = 43 = tan 37°
x = 37°
2
x = 74° $ x ก3 f(x) f(74°) = 3 sin 37° + 4 cos 37°
= 3 ⋅ 35 + 4 ⋅ 45
= 95 + 165 = 5
5.
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; θ ∈ (0, π2 ] $ sin θ cos θ = 15 tan θ ⋅ sec θ
! #&ก<=
ก$ก!ก#ก=!""!7-
กก<=#ก=!"" 23
ก"&%%!ก#
sin θ cos θ = 15 -----(1)
sin2θ + cos2θ = 1 -----(2)
*"=!ก (1) ?
sin θ = cos θ + 15 -----(3)
$ก!ก (3) 3!ก (2) (cos θ + 15 )2 + cos2θ = 1
cos2θ + 25 cos θ + 251 + cos2θ = 1
2 cos2θ + 25 cos θ 24
25 = 0
50 cos2θ + 10 cos θ 24 = 0
2 2 !ก
25 cos2θ + 5 cos θ 12 = 0
(5 cos θ + 4)(5 cos θ 3) = 0
cos θ = 45 , 35
$ θ ∈ (0, π2 ] *+#)&!#*##- cos θ = 35 2#)#
!#)!!4!+ก#
5
θ
3
ก tan θ = 43 $ sec θ = 53
*+ tan θ ⋅ sec θ = 43 ⋅ 53 = 209
6.
tan 380° !-) tan 20°
! -)ก tan 380° =
=
=
= 0.3640
tan (360° + 20°)
tan 20°
0.3640
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7.
; A $ B 37 [ π2 , π] cos2B + sin2A !#!ก#)4
! #B!
$3?'!"ก กกBก
%ก
π
%
2
(0, 1)
A, B
37#
π
( 1, 0)
*"= ?7 ; A, B 37 ( π2 , π) !!##)23
cos2B + sin2A !#4
$#)23 cos2B + sin2A !#4- π2 - π *"=ก=##
ก=##) 1 ; A = B = π2 cos2 π2 + sin2 π2 = 1
ก=##) 2 ; A = π2 , B = π cos2π + sin2 π2 = 2
ก=##) 3 ; A = π, B = π2 cos2 π2 + sin2π = 0
ก=##) 4 ; A = B = π cos2π + sin2π = 1
?4
cos2B + sin2A = 2
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8.
;$%ก!()ก& 6 ก $#ก'ก ;#%'ก()
*-#)
'กก
! -)กก!()!#*-#)ก% π !-)$%ก!#ก&
6 ก $(!#*-#)ก% π6 "#
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9.
ก23 f(x) = cos2 x2 sin2 x2 , g(x) = x + c $ 0 < x ≤ π, c &2"%ก3
; fog(c) = 1 $ 3&!%"
56ก7 g ก
! ก cos2 x2 sin2 x2 = cos x
f(x) = cos x
-)ก fog(x) = f(g(x))
= f(x + c)
=
=
=
=
=
*+ fog(c)
$ก2fog(c)
cos 2c
)-
c
cos(x + c)
cos (c + c) = cos 2c
1
1 = cos π
π
2
$
g(x) = x + π2
*"=-ก$
ก
ก. !-)$ y = 0 x = π2 )-4$ก X - ( π2 , 0)
!37 ( π2 , 0) $
ก. B"
ก
. ก5#
A
x=2
C
B
y = x + π2
? AB ก% 2 + π2 $ BC ก% 2 + π2 7ก
*+*-#)G!ก5
56ก7 g, x = 2 $$ก X !#ก%
= 12 ⋅ (2 + π2 )(2 + π2 )
= 12 ⋅ (2 + π2 )2 $
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10. H
ก()!##2!4! 45° $ 60° ก%$%!2% ;#
()'()2!4! 60° ก% 45 ! !
##ก()
! !
!#)3!#
A
45 !
60°
D
45°
B
C
*"=!#)! ADB ? AB = 45 ⋅ sin 60° = 452 3
*"=#)!#)! ABC ก??7#ก AC = AB cosec 45°
*+ AC = 452 3 ⋅ 2 = 452 6
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2
1. ก$BB()ก$ก
%;
!7
'()
กกก 7
#ก!
$%ก% 15 ! $-)ก
ก()ก#กก()ก% 5 ! ก$
#
%;&ก#)!#)!23ก"4%"4ก3(ก
(ก23 tan 18.20 = 0.3288, tan 18.30 = 0.3307, sin 18.20 = 0.3123, sin 18.30 = 0.3140)
! !
!#)3!
5 !
θ
15 !
?!4!#)$%#)4#)23*-)!3ก!#ก% arctan(0.33)
$ก!4! θ ก#%%88"
tan #) (0.3307 0.3288) = 0.0019 !4!#) 0.10
tan #) (0.3307 0.33) = 0.0007 !4!#) (0.10)(0.0007)
= 0.037
0.0019
!4! θ #)กก% 18.20 + 0.037 = 18.237
*"=ก *%#)- sin 18.237 37"9##ก%ก tan
-ก#%%88" #
!4!#) 18.30 18.20 = 0.10 sin #) 0.3140 0.3123 = 0.0017
= 0.001
!4!#) 18.30 18.237 = 0.063 sin #) (0.0017)(0.063)
0.10
sin 18.237 = 0.3123 + 0.001 = 0.3133
#)#)4#)
%;3
!7
$!ก"4%"4ก
5
= sin18.237
= 15.96 !
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ก *-)!?3ก$ก68
# %37"9#
Iกก?'()ก?B*9
3ก#ก (B!2=!= 15.81 !)
2.
- A ก- B 90 ! ก4#)- A ก-!;!?
#)"!
!4! 60° $ก4#)- B !;!?
#ก#!4! 30° กก*-ก3
! ก$ก
#;ก?!?687")
(
C
h !
60°
D
y !
A
30°
90 !
B
!!"
# (CD) h ! $ DA y !
*"=#)!#)! CDA h = y tan 60° = y 3 -----(1)
*"=#)!#)! CDB h = (y + 90) tan 30° = 1 (y + 90) -----(2)
3
? (1) = (2_ )-
1 (y + 90) = y 3
3
y = 45
DA 45 ! $3!ก (1) ก?
#ก*-"
ก% 45 3 !
3.
ก
%;
("
$()'()2!4! 10° ก%$% !-);-) x !
กJ"
2!4! 15° ก%$% !-)
% 60 !ก?;(44
"
= 4ก
%ก*- 150 ! x (ก23 sin 10° = 0.174, sin 15° = 0.259)
! H**-)2;ก=!#)ก23#
D
60 !
150 !
15°
E
C
x !
10°
F
A
B
*"=!#)! DEC *% DE = 60 sin 15° = 60 × 0.259 = 15.54
*+ EF = DF DE = 150 15.54 = 134.46 = CA
*"=#)!#)! CAB *% CA = x sin 10°
x = CA = 134.46
0.174 = 772.76 !
sin10