Sketch the given curve in polar coordinates. Do
this as follows: (a) Carefully sketch the auxiliary curve, a
rectangular graph on the
θ
,r coordinate system provided. (b) Then
translate this graph to the polar one, the x,y one.
Equation:
r = 1 + 2cos(
θ
)
(a)
r
θ
(b)
Polar
y
x
10. (5 pts.)
Write cos(cos
1
(u ) + sin
1
(v)) as an algebraic
expression containing u and v.
cos(cos
1
(u ) + sin
1
(v)) =
11. (5 pts.)
Find the exact value of cos
1
(cos(7
π
/5)).
cos
1
(cos(7
π
/5)) =
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Final Exam/MAC1114 Page 7 of 8
12. (10 pts.) To measure the height of the top of a distant
object , a surveyor takes two sightings of the top of the object
100 feet apart. The first sighting, which is nearest the object,
results in an angle of elevation of 45°. The second sighting,
which is most distant from the object, results in an angle of
elevation of 30°. If the transit used to make the sightings is 5
feet tall, what is the height of the object. You may assume the
object is on a level plane with the base of the transit.
13. (5 pts.)
Find the exact value of tan(2 tan
1
(1/4)).
tan(2 tan
1
(1/4)) =
14. (5 pts.)
Using DeMoivre’s Theorem, find all the complex
cube roots of i. [Don’t confuse cube roots with cubes!!]
15. (5 pts.)
Two sides of a triangle have lengths 3 feet and 4
feet, and the angle included between the two sides is 15°. What
is the exact length of the third side?
Final Exam/MAC1114 Page 8 of 8
16. (10 pts.) Establish the following identity. [
Hint: Begin on
the left side, expand and clean using appropriate identities, as
you work towards the right. Be very patient.
]
2sin((
α
+
β
)/2)cos((
α

β
)/2) = sin(
α
) + sin(
β
)
17. (5 pts.)
Obtain the exact value of sin(9
π
/8).
Show clearly
and neatly all the uses of appropriate identities.
sin(9
π
/8) =
18. (5 pts.)
Express the following product as a sum containing
only sines or cosines.
cos(5
θ
)cos(3
θ
) =
19. (5 pts.)
Use the Law of Sines to solve the triangle with
α
= 105°,
γ
= 35°, and c = 24. You may assume that the standard
labelling scheme is used.
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 Spring '08
 Storfer
 Trigonometry, pts

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