Precalculus Midterm Solutions
1.
11.
21.
31.
41.
b
d
c
b
a
2. d
12. d
22. c
32. b
42. b
3
(360) = 270.
4
1. b.
!
3.
13.
23.
33.
43.
a
d
c
b
c
4.
14.
24.
34.
44.
2. d. sec" =
b
d
b
b
b
r
x
d
d
d
a
c
6.
16.
26.
36.
46.
c
b
d
c
a
7.
17.
27.
37.
47.
b
c
d
a
d
8.
18.
28.
38.
48.
480° " co - terminal with 120°
x < 0, y > 0 - choices b and c
Co - terminal - terminal
side the same as given angle.
b
d
d
d
b
9.
19.
29.
39.
49.
c
d
b
b
a
10. c
20. b
30. c
40. d
50. a
4. b
3. a
!
5. d.
5.
15.
25.
35.
45.
6. c tan120° = " 3
choice b has tan# =
3
2
"1
2
2
tan 45° = 1
2
7. b.
2
" 2+2
"
+1=
2
2
=" 3
sin225° = "
!
!
!
8. b.
sin t =
!
" 8 "2 2
=
3
3
!
a. cos " = #1...possible
125
b. tan " =
...possible
4
5" 180°
#
= 112.5° 10. c.
9. c.
99
8
"
c. sec " =
...impossible (sec " $ 1)
100
1
d. cot " = ...possible
12
!
cos" and sec " are negative.
12. D.
5
12
tan " = ,cot " =
12
5
11. d.
all have zero
in the denominator
!
!
Heading =
360° " 28°15# = 331°45#
13. d.
!
!
14. d.
4 in
makes no sense
radian
cos2" = cos2 " # sin 2 "
15. d.
$ #3 ' 2 $ 4 ' 2 #7
cos2" = & ) # & ) =
% 5 ( % 5 ( 25
a. pd. = 360°
! = 360° = 720°
b. pd.
1
c. pd. =
d. pd. =
360°
= 270°
4
3
360°
3
4
cos65°cos12° + sin65°sin12° = cos(65° "12°) = cos53°
!
2
17. c.
cos Acos B + sin Asin B = cos( A " B)
16. b.
18. d.
360°
=2
180°
19. d a) negative cosine curve with no shift
b) positive cosine curve shifted 90° right
c) positive sine curve shifted 45° right
d = 1,a = 2,b =
d = 1,a = 3
range[ d " a,d + a] = ["2,4 ]
= 480°
!
!
d = 0,a = 3,c = 0
!
20. b. y = 2 " 2sin
2
( x + 45°)
3
21. c. period = "...b =
y = 3cos2x
!
sin 2 x + cos 2 x = 1
2"
= 2 22. c.
"
" sin x %
cos x
= cos x tan x = cos x$
' = sin x
# cos x &
cot x
" cos x %" cos x %
1
csc x ( cot x sin x =
($
'$
'
sin x # sin x &# 1 &
1( cos2 x sin 2 x
=
=
= sin x
sin x
sin x
sin2x 2sin x cos x
=
= 2cos x
sin x
sin x
!
sin 4 x cos 4 x
1
! Choice a, c, d are angular velocities.
"
"
=1
24.b.
4
4
Choice b is a linear velocity.
1
sin x cos x
2
2
(tan x # sec x )(tan x + sec x ) = tan x # sec x = #1
23. c. sin 4 x cot 4 x sec 4 x =
You are
!given SSS so this is
!
You are given SAS so this is
25. d.
a law of cosines problem.
!
normally a law of cosines problem.
26. d. But the triangle is impossible because
a + b < c and in any triangle, the sum of
any two side must be greater than the third.
2sin 2 x + sin x "1 = 0
!
(2sin x "1)(sin x + 1) = 0
1
1
2sin x = 1 # sin x =
2
2
27. d. ASTC # x = 60° in quad. I and IV 28. d ASTC " x = 30° in quad. I and II
x = 60° or x = 300°
x = 30° or x = 150°
2cos x = 1 " cos x =
sin x + 1 = 0 # sin x = "1 # x = 270°
!
cos A = cos B is obviously true when A = B
Attempt to draw it. You
But because of ASTC, there are 2 quadrants
29. b.
30. c. should be convinced that
when cosine values are the same. For instance,
the triangle is impossible.
cos60° = cos 300°.
!
!
31. b.
32. b.
33. b. tan28° =
x
" x = 75tan28°
75
4
# " = 34.8°
7
2
b) cos " = # " = 66.4° !
5
34. b.
35. a.
6
c) tan " = # " = 56.3°
4
8
d) sin " = # " = 53.1°
10
a) sin " =
BC
7
BC = 7sin20° = 2.39
sin20° =
!
By Pythag. Thm, other leg is 5
5
12
5
sin A =
or cos A =
or tan A =
13
13
12
A = 22.6°
!
36. c.
!
tan 33°40" =
37. a.
600
x
cos 46°10" =
600
x=
= 900.8
33°40"
!
38. d.
AB =
18
AB
18
= 25.99
46°10"
!
25
25
"x=
= 65.13
x
tan21°
40. d.
y
tan 31° =
" y = 64.13 ft
65.13
tan21° =
39. b.
318
" P = 57.8°
200
270 + 57.8 = 327.8°
tan P =
!
v = r"
s = r"
41. a. v = 3 in # 180° # 2$ = 3$ in 42. b.
sec 360° sec
!
% 2$ (
s = 6.5 # 55°'
* = 6.2
& 360° )
a
b
=
" asin B = bsin A
sin A sin B
s
s = r" # " =
$ bsin A '
r
44. b. B = sin#1&
) = 23°
% a (
8.4 cm 360°
"=
$
= 133.7°
3.6 cm 2%
C = 180 #102 # 23 = 55°
!
43.c.
a
b
c 2 = a 2 + b 2 " 2abcosC
=
" asin B = bsin A
sin A sin
! B
45. c.
46.a. c = !a 2 + b 2 " 2abcosC
asin B
b=
= 8.9
c = 5.05
sin A
c 2 = a 2 + b 2 " 2abcosC
!
s=
a+b+c
=9
2
!
a2 + b2 " c 2
47. d. cosC =
48. b. Area = s( s " a)( s " b)( s " c )
2ab
Area = 9( 4 )(2)( 3) = 14.7
# 120 2 + 197 2 " 85 2 &
C = cos"1%
( = 13.4°
$ 2(120)(197) '
!
!
C = 180 " 22 " 44 = 114°
c
b
c sin B
=
#b=
= 1520.7
sinC sin B
sinC
CD
sin22° =
# CD = 570
1520.7
49. a.
Switch to radian mode
d = 12,a = 7,c = 19
2#
50. a. period = 20 " b =
20
%2#
(
p = 12 + 7cos' ( t $19)*
& 20
)
!
!