Download MATH 026 FINAL EXAM, SAMPLE C

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MATH 026
FINAL EXAM, SAMPLE C
4. How far up the side of building will a 3-meter long ladder reach if
the foot of the ladder is 1 meter from the base of the building?
1. Find the equation of the circle with the graph below.
-5
-4
-3
-2
-1
0
1
2
3
4
a) 15 meter
6
7
8
b) 2 meters
√
c)
2 meters
√
d) 2 2 meters
√
e)
10 meters
-1
-2
-3
5. Find the area of a sector of a circle of radius 10 inches intercepted
π
.
by a central angle of θ =
10
-4
a) 10π square inches
a) (x − 1)2 + (y + 2)2 = 5
b) 20π square inches
b) (x + 1)2 + (y + 2)2 = 5
c) 2π square inches
c) (x − 1)2 + (y − 2)2 = 5
d) 5π square inches
d) (x − 1)2 + (y + 2)2 = 5
e)
π
square inches
2
e) (x + 1)2 + (y + 2)2 = 0
2. Find the angle of least positive measure that is coterminal with the
29π
angle −
.
4
π
a)
4
b)
3π
4
c)
5π
4
d)
7π
4
e)
6. A windmill with a blade 12 feet long rotates at an angular verlocity
of 45◦ per second. What is the linear velocity of the tip of the
windmill blade?
a) 2π feet per second
b) 4π feet per second
c) 8π feet per second
d) 3π feet per second
e) 24π feet per second
7. Find the value of
9π
4
a) −1
3. For an angle θ whose terminal side lies in Quadrant III, which of the
following quantities is always positive?
b) 1
c)
a) cos θ
1
2
b) csc θ tan θ
d) −
c) sec θ sin θ
e) 0
cos θ
d)
sec2 θ
e) csc θ cot θ
1
1
2
sin 35◦
cos 55◦
MATH 026
FINAL EXAM, SAMPLE C
11π
8. Give the exact value of sec −
.
3
12. Find the inverse function of f (x) =
2
a) √
3
a) f −1 (x) =
2x + 1
x−3
2
b) − √
3
b) f −1 (x) =
x−3
2x − 1
c) −2
c) f −1 (x) =
2x + 3
2x + 1
d) f −1 (x) =
x+3
2x + 1
e) f −1 (x) =
2x − 1
x+3
d) 2
1
e) −
2
−x + 3
.
2x − 1
9. Find the point on the unit circle that corresponds to the angle θ =
5π
√
.
13. Find the exact value of sec(tan−1 ( 3)).
6
√ !
√
1
3
3
a)
,
a)
2 2
2
√
b)
3 1
,
2 2
c)
d)
−
√
3 1
,
2 2
!
3
1
,−
2
2
!
√
1
2
√
2 3
d)
3
c)
√ !
3
1
− ,
2 2
e)
b) 2
!
e) The value does not exist.
14. Simplify sin x sec x + cos x csc x.
a) csc x sec x
10. Give the amplitude, period and phase shift of the function f (x) =
−3 sin(2x + π).
b) −2 sec x tan x
c) cot x sec x
a) Amplitude 3, Period π, Phase shift −π
d) − cot x cos x
π
b) Amplitude 3, Period 4π, Phase shift −
2
π
c) Amplitude 3, Period π, Phase shift −
2
π
d) Amplitude −3, Period π, Phase shift
2
e) 1
15. Give the exact value of cos(150◦ ) cos(15◦ ) + sin(150◦ ) sin(15◦ )
a) 1
e) Amplitude 3, Period 2π, Phase shift π
b) 0
11. Which one
x of the following is a vertical asymptote of the graph for
y = sec
?
3
a) x = 0
c) x = 2π
π
2
e) x =
3π
2
1
d) − √
2
e) −1
b) x = π
d) x =
1
c) √
2
2
MATH 026
FINAL EXAM, SAMPLE C
16. The terminal side of an angle θ lies in quadrant II and sin θ =
Find sin 2θ.
1
3
. 20. Which of the following is a solution to the equation 2 sin θ cos θ = ?
5
2
π
12
π
b)
6
π
c)
3
a)
3
a)
5
b)
12
25
c)
24
25
d) −
12
25
e) −
24
25
17. Evaluate tan
3π
12
e)
2π
3
21. If A = 30◦ , a = 20, B = 45◦ , find b.
π
.
12
a) 20
√
a)
d)
b) 30
√
c) 10 2
√
d) 20 2
√
e) 40 2
3
2
√
2− 3
2
√
c) 2 − 3
√
d) 2 + 3
√
e)
3−2
b)
18. Convert the following to an expression involving a product of trigonometric functions:
cos(6x) − cos(10x)
a) sin(4x) sin(2x)
b) 2 cos(8x) sin(4x)
c) 2 sin(6x) sin(4x)
22. (For Problems 22 and 23, √
refer to the diagram for Problem 21.) If
A = 30◦ , a = 5, and b = 5 2, which is a possible value for B?
d) 2 sin(8x) sin(2x)
e) 2 sin(6x) sin(2x)
a) 90◦
19. Determine the exact value of the expression cos
a)
1
1
√ +
2
2 2
b)
1
√
2 2
c)
1
1
√ −
2
2 2
b) 120◦
5π
π
cos .
8
8
c) 60◦
d) 135◦
e) 150◦
23. Find the angle A in the triangle 4ABC if a =
π
6
π
b)
4
π
c)
3
1
1
d) − √ +
2
2 2
a)
1
e) − √
2 2
3
d)
2π
3
e)
3π
4
√
19, b = 2 and c = 3.
MATH 026
FINAL EXAM, SAMPLE C
24. A hinge is open at an angle of 60◦ . If one of the arms of the hinge is
1 inch long and the other arm is 3 inches long, what is the distance
between the ends of the arms?
a)
√
7
Trigonometric Identities
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
b) 7
√
13
c)
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
d) 13
√
10
e)
tan α + tan β
1 − tan α tan β
tan α − tan β
tan(α − β) =
1 + tan α tan β
tan(α + β) =
25. Determine the area of the equilateral triangle where all sides have
length 4.
r
α
1 − cos α
=±
2
2
r
α
1 + cos α
cos = ±
2
2
1 − cos α
sin α
α
=
tan =
2
sin α
1 + cos α
√
sin
a) 4 3
b) 48
√
c) 32 6
√
d) 16 2
√
e) 2 6
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1. B
2. B
3. C
4. D
5. D
6. D
7. B
8. D
9. D
10. C
11. E
12. D
13. B
14. A
15. D
16. E
17. C
18. D
19. E
20. A
21. D
22. D
23. D
24. A
25. A
1 + cot2 θ = csc2 θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
tan 2θ =
2 tan θ
1 − tan2 θ
1
[sin(x + y) + sin(x − y)]
2
1
cos x sin y = [sin(x + y) − sin(x − y)]
2
1
cos x cos y = [cos(x + y) + cos(x − y)]
2
1
sin x sin y = [cos(x − y) − cos(x + y)]
2
sin x cos y =
x+y
x−y
cos
2
2
x+y
x−y
sin
sin x − sin y = 2 cos
2
2
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
x+y
x−y
cos x − cos y = −2 sin
sin
2
2
sin x + sin y = 2 sin
4