Download Exam Review #2

Advanced Math (1 )
IMPORTANT!!!!!
Semester Exam Review #2
SOLUTIONS
Name _______________
PLEASE LET YOUR TEACHER KNOW IF YOU FIND A TYPO!
1. Find the value of x to the nearest yard. 53 YARDS
˙
45
cos32° =
x
x
45
x=
cos32°
32°
x = 53 yards
45 yards
2. In !ABC, "A = 44° , "B = 65° , and a = 4 . Find side b to the nearest tenth. b=5.2
sin44° sin 65°
=
4
b
4sin65°
b=
= 5.2
sin 44°
4
44°
65°
B
A
3. In !ABC, "A = 57° , a = 8, and b = 7 . Find the number of solutions to the triangle. ONE
sin57° sin B
=
8
7
7sin 57°
B = sin !1
= 47.2° or 132.8°
8
138.2° + 57° !180° so only 1 solution.
8
7
57°
B
A
4. The angle of depression from the top of a cliff 750 m high to the base of a streetlight is 68 ° .
To the nearest meter how far is the street light from the foot of the cliff? 303 feet
5. Does csc x = csc (-x) for all values of x? NO
(Q1 and Q4 don’t have same sign for csc (or sin).)
1
6. Find the equation of the graph below. y = 2sinx + 1
π
3π
2π
7. Find the equation of the graph below. y=2cos[(2π/3)(x–1)]–1
#
8. Graph the equation y = "2sin (x " 3) + 1.
2
!
9. Expand and simplify:
=
sin 2x
1! cos 2x
2 sin x cos x
(
1! cos2 x ! sin2 x
)
2 sin x cos x
1! cos2 x + sin2 x
2 sin x cos x
=
sin2 x + sin2 x
2 sin x cos x
=
2 sin2 x
cos x
=
sin x
= cot x
=
2
10. What quadrant is x in if cos x < 0 and sin x < 0? Q3
!
11. Evaluate csc . 2
6
#
"&
12. Graph y = 2 ! cos% x + ( .
$
3'
"
#
3
5"
3
14. Convert
4!
to degrees. 144°
5
!
!
15. Find the length of side BC to the nearest tenth. 12.7 or 2.27
A
14
15
B
14 2 = 15 2 + x 2 " 2 #15 # x # cos60°
60°
15
0 = x 2 "15x + 29
x = 12.7, or x = 2.27
C
x
!
#
"&
!
16. What is the horizontal shift of the graph of y = !3cos% x ! ( ? Answer: to the right
$
2'
2
17. Find the equation that matches the graph below: y = –3 cos 2x
π
3π
2π
3
18. Find cos! if θ is an angle in standard position with its terminal ray passing through the
!2
5
=!
point (–2, 4). cos x =
5
2 5
(–2, 4)
θ
19. cot C = _____
cot C =
A
adj 2 21
21
=
=
opp
4
2
10
4
C
B
2
2
10 ! 4 = 84 = 2 21
20. If sec x = 2 find all values of x to the nearest degree for 0° ! x < 360° . x = 60°, 300°
21. If sec x is positive, in what quadrants could x be? Quadrants I or IV.
22. If tan x = 3 find all values of x to the nearest degree for 0° ! x < 360° . x = 60°, 240°
"! %
23. Graph a full period of the following function. y = 2sin$ x '
#2 &
2
-2
4
24. A 150 ft building casts a shadow of 250 feet at 3:00 pm. What is the angle of elevation from
the end of the shadow to the top of the building? θ = 31°
4
3
and sin θ > 0. Find tan θ. tan θ = !
5
4
6! 2
26. Find the exact value of cos 75 ° . cos 75° =
4
25. cos! = "
27. Find the exact value of sin 42 ° cos 12 ° - cos 42 ° sin 12 ° . sin (42° – 12°) = sin 30° = 0.5
28. Derive the formula for sin 2a from the sin (a + a) formula.
sin 2a =
sin (a + a)
=
sin a cos a + cos a sin a
=
2 sin a cos a
29. Derive the formula for cos 2a from the cos (a+a) formula.
cos 2a =
cos (a+a)
=
cos a cos a – sin a sin a
=
cos2 a – sin2 a
30. Solve the following equations for ALL values of x in radians in the interval 0 ! x < 2"
a
b)
# 3"
7" &
tanx sinx + sin x = 0 x = $0,
, ",
' SET = 0, FACTOR
4
4(
%
sinx ( tanx + 1) = 0
sinx = 0 or tanx + 1 = 0
!
" 7! 11! 19! 23! %
1
sin 4x cos 2x - cos 4x sin 2x = !
x=# ,
,
,
& spin TWICE
2
$ 12 12 12
12 '
This simplifies using the sin(a–b) formula:
1
sin(4x ! 2x) = !
2
1
sin2x = !
2
7" 11" 19" 23"
2x =
,
,
,
6 6
6
6
7" 11" 19" 23"
x=
,
,
,
12 12 12 12
5
c)
4 sin2 x - 1 = 0
d)
3 sin x = 4
sin2 x = 1/4
sin x = ±1/2 PLUS OR MINUS!
% ! 5! 7! 11! "
x=& ,
,
,
#
6
6 $
'6 6
sin x = 4/3 NO SOLUTION!
31. Find ALL possible angles and sides.
A
12
A1 = 86.5°, a1 = 18.6, C1 = 53.5°
A2 = 13.5°, a2 = 4.4, C2 = 126.5°
REMEMBER sin-1x has solutions in QI and QII.
15
C
40°
B
32. Find the missing angles.
A
7
C
3
B
A = 123.2°, B = 40.3°, C = 16.5°
9
°
°
33. Solve each equation on the interval 0 ! x < 360.
a) 3 csc x - 5 = 0
2
b) 2 sin x - sin x - 1 = 0
2
c) tan x - 1 = 0
csc x = 5/3
sin x = 3/5
x = {36.9°, 143.1°}
(2 sin x + 1 )( sin x – 1) = 0
sin x = –.5, sin x = 1
x = {90°, 210°, 330°}
(tan x + 1) ( tan x – 1) = 0
tan x = –1 or tan x = 1
x= {45°, 135°, 225°, 315°}
6
34. Verify the identity:
cos "
1+ sin "
+
= 2sec "
1+ sin "
cos"
cos" (1# sin " )
!
35. Verify the identity:
!
1+ sin "
=
(1+ sin " )(1# sin " )
cos "
cos" (1# sin " ) 1+ sin "
+
=
cos"
1# sin 2 "
cos" (1# sin " ) 1+ sin "
+
=
cos"
cos2 "
1# sin " + 1+ sin "
=
cos "
2
=
cos"
2sec " =
+
cot 2 "
+ sin " csc " = csc "
1+ csc "
cot 2 !
+1=
1+ csc
!!
cot 2 ! +1+ csc !
=
1+ csc !
csc 2 ! + csc !
=
1+ csc !
csc ! csc ! +1
=
1+ csc !
csc ! = csc ! !
(
)
36. A boat sails from port on a course N30 ° W for 75 miles and then turns to a SW course for
250 miles.
a) How far must the boat sail to return to port? 241.7 miles
b) What course must the boat sail to return to port?
7
N 62.44° E
37. Label the points whose coordinates are:
a) A(4, -240°)
b) B(-5, 135°)
c) C (3, 300°)
d) D(-1, -150°)
A
D
C
B
38. Convert to rectangular coordinates.
a) (5, -225°)
b) (-4, 150°)
" 5 2 5 2%
$!
,
'
2 &
# 2
(2
c) (3, 270°)
3,!2
)
(0,!3)
39. Convert to polar coordinates. Round to the nearest degree.
7 3 7
!
a)
(-4, -1)
b)
( 2 , 2 )
c)
( 17,194°)
(-7, 0)
(7,330°)
(7,180°)
40. Find four polar coordinates for the point (2, –40°).
a)
41.
(2, –
)
b)
(2, +
)
(2, – 400)
b)
(2, + 320°)
c)
(–2 , –
c)
(–2 , – 220°) d)
Find the equation of the following graph. y = sec x
8
)
d)
(–2 , + )
(–2 , +140°)
42.
Read the following carefully, then fill in the blank.
a)
If tan θ = k, then tan (–θ) = ____–k _______
b)
If sec θ = k, then sec (–θ) = _____ k _______
c)
If cos θ = r, then cos (θ + 2π)= ___ r _____.
d)
If tan θ = r, then tan (θ + π)= ____ r ____.
43.
Find the smaller angle made between the two hands of a clock at 4:57.
120°+6°⋅3+
57
" 30° =166.5°
60
!
44.
The minute hand of a clock is 9 inches long. How far (in inches) has the tip of the minute
hand moved (around in a circle) in 25 minutes?
25
" 2# " 9 =7.5π
60
45. ! Planet Moog is 2.8 x 107 km from Earth and has an apparent size of about 0.001°. What
is the approximate diameter of planet Moog? You don't need to evaluate your expression for this
diameter.
.001
" 2# " 2.8 "10 7 = 488.7 km
360
46.
a)
Convert the following.
! 42.21° to degrees, minutes, and seconds.
42° 0.21⋅60’ = 42° 12.6’ = 42° + 12’ + 0.6⋅60’’=42° 12’ 36’’
b)
76°6’36’’ to decimal degrees.
6
36
76+ +
= 76.11
60 3600
!
9