COMPARISON OF 2012 AND 2015 MATH 1060 FINAL EXAMS
2015
2012
(1) Find the exact value of the following or state
that it is undefined:
(a) sin( 3π
4 )
(b) sec(− 5π
6 )
(c) csc(720◦ )
(d) tan(150◦ )
(1) Find the exact value of each of the following
or state that it is undefined.
(a) tan(30◦ )
(b) csc(240◦ )
(c) cot(180◦ )
(d) sec( 15π
4 )
(e) cos(− 2π
3 )
(f) sin(− π4 )
(2) Find the exact value of each expression in
radians.
√
(a) arccos(− 22 )
(5) Evaluate each expression.
(a) cos−1
(− 21 ) (b)
(c)
√
arctan( −3 3 )
√
arccsc( 2 3 3 )
(b) tan sin−1 −
√
3
2
.
(3) Find an equivalent algebraic expression for
tan(arcsin(x)).
(6) Find an equivalent algebraic expression for
the composition. sec(sin−1 x).
(4) Use appropriate identities to find the exact
5
π).
value of the expression sin( 12
(3) (a) Complete the formula. cos(α − β) =
(b) Use the formula
from a) to find the exact
π
=
value. cos
12
√
(5) Let z = −2 3 + 2i.
(a) Find the trigonometric form of z. Use
degree measure.
(b) Use the trigonometric form to find the
square roots of z.
(6) Let z = 36(cos( 18 π) + i sin( 18 π)) and w =
9(cos( 38 π) + i sin( 38 π)). Write the expression
z
in the form a + bi. Give exact values, not
w
decimal approximations.
1
(11) (a) Write the complex number (−1 − 2i) in
trigonometric form. Write your answer
in degrees rounded to the nearest tenth.
(b) Use De Moivre’s theorem to find (−1 −
2i)6 .Write your answer in the form a +
bi.
√
(18) Find all of the fourth roots of −8 + 8i 3.
Write your answer as r(cos θ + i sin θ).
(17) Multiply. 3(cos 20◦ + i sin 20◦ ) · 2(cos 25◦ +
i sin 25◦ ) Give the exact answer in the form
a + bi.
(7) Find all real numbers x (in radians) that
satisfy the equation sin(2x) + cos(2x) = 0.
(4) Find all exact real numbers in radians in
the interval√[0, 2π) that satisfy the equation.
2 sin(2x) − 3 = 0
(8) Consider the function y = 3 cos(2x + π2 ) − 1.
(a) Find the following quantities. Phase
shift, Period, Amplitude, Range
(b) Graph the function and give the coordinates of the five key points on the
graph.
πx π
.
+
(2) Consider the function y = 2 cos
4
2
(a) Find the phase shift.
(b) Find the period.
(c) Find the frequency.
(d) Find the amplitude.
(e) Find the range.
(f) Graph the function and give the coordinates of the five key points on the
graph.
(9) Find the exact value of sin(α − β) if α is in
quadrant III with sin(α) = − 23 and β is in
quadrant I with cos(β) = 35 .
(10) Find the exact value of the following expression and simplify your answer: 1 −
2 sin2 (22.5◦ ). Hint: You will need to use
an identity.
(12) Find the area of a triangle with the lengths
of the sides 220m, 234m, and 160m. Round
to the nearest integer.
(11) The Santa Monica Ferris Wheel has 20 gondolas and a diameter of 90 feet. Please answer
the following. Give exact values, not decimal
approximations.
(a) What is the angle in radians between
successive gondolas?
(b) When the ferris wheel is turning, the
gondolas pass by the entry gate at a rate
of 1 gondola every 1.6 seconds. What
is the ferris wheel’s angular velocity in
radians per second?
(c) In the movie 1941, the ferris wheel broke
free and rolled down the Santa Monica
pier. Assuming it continued to turn at
the same angular velocity, how fast was
the wheel moving down the pier in feet
per second?
(10) Suppose the tip of a lawn mower blade of radius 11 inches long is spinning 2,700 rev/min.
Find the linear velocity in miles per hour.
Round to the nearest tenth. Given that 1
mile = 5,280 feet.
(19) The hour hand of a clock is 5 inches long.
Find the distance in inches that the tip will
move in 10 hours. Give an exact answer.
(22) The radius of each of the tires on Shane’s car
is 14 inches long. Find the angular velocity
in radians per hour of a point on the outside
edge of a tire when his car is going 65 miles
per hour. Round to the nearest tenth. Given
that 1 mile = 5,280 feet.
(12) Write an equivalent rectangular equation for
the polar equation r = 9 cos(θ).
(13) If a force of 150 pounds is required to push
a piano up a ramp that is inclined 25◦ ,then
what is the weight of the piano? You must
draw a sketch to show the given situation.
Round to the nearest integer.
(13) A small plane is traveling with an air speed
of 200 mph at a bearing of 45◦ . If a 30 mph
wind is blowing at a bearing of 120◦ , then
what is the ground speed and true bearing of
the plane? Round to 1 decimal place. Your
work must include a sketch showing the given
situation.
(21) Two prospectors are pulling on ropes attached around the neck of a donkey that
does not want to move. One prospector pulls
with a force of 55 lb, and the other pulls
with a force of 80 lb. If the angle between
the ropes is 25◦ , how much force must the
donkey use in order to stay put? (Assume
the donkey knows the proper direction in
which to apply his force.) You must draw a
sketch to show the given situation. Round
to the nearest tenth.
(14) Complete the table and graph the polar equation r = 2 + 2 sin(θ).
(14) Graph the polar equation r = 2 cos 2θ. State
at least four exact (r, θ) points on the graph.
(15) Verify
the
following
identity
1 + sin(x) cos(x) − tan(x)
+
= 2 csc(2x)
cos(x)
sin(x)
(16) Prove that the equation is an identity.
sin 3x = sin x(3 cos2 x − sin2 x)
(7) Prove that the equation is an identity.
1 − sin2 x
cot(−x) =
cos(−x) sin(−x)
(16) Eliminate the parameter and identify the
graph
√ of the parametric equations x = 2t − 3,
y = t − 2. Draw a sketch showing at least
three exact points on the graph.
(20) Eliminate the parameter and identify the
graph of the pair of parametric equations:
x = 2 sin t cos t and y = 13 sin 2t
(17) Two sides and the included angle of triangle
4ABC are given:
a = 5, b = 8, γ =
∠BCA = 40◦ . Solve the triangle and give all
side lengths and angle degrees to 2 decimal
places.
(15) Assume a = 7, b = 9, and c = 12. Solve the
triangle. Round to the nearest tenth.
(18) Let v = h2, −3i and w = h−1, 5i
(a) Find 2v − w.
(b) Determine the magnitudes of v and w.
(c) Determine the direction angle of w.
Round to the nearest 10th of a degree.
(d) Determine the cosine of the angle between v and w.
(8) Use the vectors u = h−4, 5i and v = h6, −7i
and write answers in the form ha, bi
(a) 2u − v.
(b) Find |v|.
(c) Find u · v.
(19) Two sides and an angle of triangle 4ABC are
given:
a = 6, b = 8, α = ∠CAB = 40◦ .
Solve the triangle and give all side lengths
and angle degrees to 2 decimal places.
(9) Solve
swers
C
the triangle.
to
the
Round annearest
tenth.
γ
17.2
11.4
125◦
β
A
c
B