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MA 15400
1)
2)
Exam 2A
Summer 2009
Given ∆ABC with ∠C = 90°, c = 215.1, and α = 49 42′ ; approximate b to the
nearest tenth and ∠B or β to the nearest minute. Hint: Draw a triangle.
A
b = 182.4, ∠B = 4018′
B
b = 139.9, ∠B = 4058′
C
b = 164.0, ∠B = 4018′
D
b = 164.0, ∠B = 40 58′
E
b = 139.1, ∠B = 4018′
∠B = 90 − 49 42′ = 4018′
B
b
215.1
b = 215.1cos(49.7 )
C
cos 49.7 =
49.7°
b
b = 139.1
A measure the height of a large billboard in Times Square in New York City, two
sightings 500 feet along a street from below the billboard are taken. The angle of
elevation to the bottom of the billboard is 55° and the angle of elevation to the top
of the billboard is 61°30'. What is the height of the billboard to the nearest foot?
h−b
500
500 tan ( 55 ) = h − b
tan ( 55 ) =
A
207 ft.
B
79 ft.
C
D
30 ft.
219 ft.
E
None of the Above.
b
h
tan ( 61.5 ) =
h
500
500 tan(61.5 ) = h
61.5°
55°
500 tan(55) = 500 tan(61.5) − b (substitution)
b = 500(tan(61.5) − tan(55))
500
b = 206.8
3)
Ship A leaves port at 2:00 PM and sails in the direction S 23° E at a rate of 25
miles per hour. The second ship, B, leaves the port at 2:30 PM and sails in the
direction N 67° E at a rate of 30 miles per hour. What is the heading from ship A
to ship B at 4:00 PM?
A
S 19 W
B
N 23 E
C
S 48 W
D
N 19 E
E
N 42 E
heading = θ
∠A − 23 = θ
Port
67°
45
tan A =
23°
50
θ
A
1
45
= 0.9
50
A = 42
θ = 42 − 23 = 19
N 19 E
A
MA 15400
4)
5)
Exam 2A
A man and his son are playing paintball. The man is on the roof of a building and
wants to hit his son who is standing on the ground. The angle of depression from
the man’s paint gun to his son’s feet is 65° and the height from the base of the
building to the man's paint gun is 28 feet. How far will the paint ball travel
through the air from the paint gun until it hits the feet of the son? Assume the
paint ball travels in a straight line. Round to the nearest tenth of a foot.
A
66.3 ft.
B
25.4 ft.
C
D
60.0 ft.
30.9 ft.
E
None of the Above.
65°
cos(25 ) =
28
d
28
d
28
cos(25 )
d = 30.9 ft.
d=
Find all solutions of the equation below using n as an arbitrary integers.
4 cos 2 α − 1 = 0
A
B
C
D
E
6)
Summer 2009
π
2π
α = + π n,
+π n
3
3
5π
π
+π n
α = + π n,
6
6
2π
π
α = + 2π n,
+ 2π n
3
3
π
5π
α = + 2π n,
+ 2π n
6
6
None of the Above.
α will be in all 4
quadrants with a
reference angle
of
4 cos 2 α = 1
π
3
1
4
1
cos α = ±
2
cos 2 α =
α=
π
3
+ π n and α =
2π
+πn
3
Find all solutions in the interval [0, 2π) for the equation below. How many
solutions are there?
π
Reference angle is
in Q II and Q IV(tan is
6
π
1

tan  2 x −  = −
negative). Terminal sides are in opposite
2
3

5π
directions. Use
+ π n equal to angle.
6
A 2
π 5π
B 3
2x − =
+πn
2
6
n
x
C
4
-1
π/6
4π
2x =
+πn
D 5
4 solutions
0
2
π/3
3
E 6
1
7π/6
2π π
x=
+ n
2
5π/3
3 2
2
MA 15400
7)
A
cot(23 40′) = tan(66 20′)
B
π

π 
π 
sin  − θ  = sin   cos θ − cos   sin θ
3

3
3
C
tan ( 55 ) =
E
tan ( 25 ) − tan ( 30 )
1 + tan(25 ) tan(30 )
The false one is the one with
tangent or 55°. The numerator
should have a + and the
denominator a – to be the sum of
two angles tangent formula.
π 
π 
π 
sin   = 2sin   cos  
3
6
6
π
 5π 
arcsin  sin
=−
4 
4

Find the exact value of sin(120 − 225 ) .
A
B
C
D
E
9)
Summer 2009
Which statement is false? Use your knowledge of the sum/difference formulas,
double-angle formulas, cofunction formulas, and/or inverse trig functions.
D
8)
Exam 2A
− 6+ 2
4
− 6− 2
4
− 3 −3
4
6− 2
4
6+ 2
4
= sin(120) cos(225) − cos(120) sin(225)
Remember: The sin is positive in Q II and negative
in Q III; the cos in negative in both these quadrants.
 3 
2   1 
2
= 
  −
 −  −   −

 2  2   2  2 
=−
6
2 − 6− 2
−
=
4
4
4
If α and β are acute angles such that sin α =
value of cos(α − β ) .
A
B
C
D
E
253
325
204
325
36
−
325
323
325
None of the Above.
7
12
and cos β = , find the exact
25
13
25
7
α
24
= cos α cos β + sin α sin β
 24  12   7  5 
=    +   
 25  13   25  13 
288 35 323
=
+
=
325 325 325
3
13
β
12
5
MA 15400
10)
Exam 2A
Summer 2009
Find the exact solutions of the equation below in the interval [0, 2π) .
sin t − sin(2t ) = 0
A
B
C
D
E
11)
t = 0,
t=
π
3
, π,
4π
3
π π 3π 5π
, ,
,
3 2 2 3
5π
π
t = 0, , π ,
3
3
π 5π
t= ,
3 3
None of the Above.
sin t − 2 sin t cos t = 0
sin t (1 − 2 cos t ) = 0
sin t = 0 1 − 2 cos t = 0
cos t =
t = 0, π
π 5π
3
,
3
If a projectile is fired from ground level with an initial velocity of v feet per
second and at an angle of θ degrees with the horizontal, the range R of the
v2
projectile is given by R = sin θ cos θ . If v = 90 feet per second, approximate
16
the angle(s) that result(s) in a range of 200 feet. Round to the nearest tenth of a
degree.
902
sin θ cos θ
16
4050
200 =
(2)sin θ cos θ
16
0.790123457 = sin(2θ )
200 =
A
θ = 52.2
B
θ = 26.1 , 63.9 C
θ = 11.6 , 78.4 D
E
θ = 23.3
None of the Above.
2θ = 52.197
θ = 26.1
12)
t=
(-1,0)
1
2
The expression
A
B
C
D
E
2sin 2 x
3
sin(4 x)
3cos 2 x
2sin x
3cos x
4sin x
3
4sin 2 x
3
2θ = 127.803
θ = 63.9
sin 2 (2 x)
is equivalent to which of the following?
3cos 2 x
sin 2 ( 2 x ) (2 sin x cos x)2
=
3cos 2 x
3cos 2 x
4sin 2 x cos 2 x
=
3cos 2 x
4sin 2 x
=
3
4
(1,0)
MA 15400
13)
Summer 2009
  7π
Find the exact value of the following: sin −1 cos
  6
A
B
C
14)
Exam 2A
4π
3
π
  7π
sin −1  cos 
  6
D
−
E
−

3

−1

  = sin  −

 2 
=−
3
5π
3

 .

π
(Inverse sin returns a
3
negative value as a negative
angle in Q IV.)
π
6
π
3
Use the quadratic formula (and an inverse trigonometric function) to approximate
the solutions of the following equation in the interval [0, 2π ) to three decimal
places. Check mode on calculator.
3sin 2 x + 4sin x − 1 = 0
A
x = 12.430
B
x = 0.217, 2.925
C
D
x = 3.359, 6.066
No solution.
E
None of the Above.
−4 ± 16 − 4(3)(−1) −4 ± 28
=
2(3)
6
sin x = 0.21525
sin x = −1.548
xR = 0.216948
(This is impossible)
sin x =
x = 0.217
x = π − xR = 2.925
5
MA 15400
15)
Exam 2A
Summer 2009
1 
Which is the graph of y = cos −1  x  ?
2 
The domain of the
function has been
‘doubled’. The domain is
[-2, 2]. The range of an
inverse cosine function is
[0, π]. The graph that fits
on this form of the exam
is C.
6