Math 127 Final Review Sheet
Graphing angles and radian measure (4.1):
1.
Jamie drew a circle. Next, she drew 20 lines radiating from the center to split the circle into equal sectors. Then, she
graphed an angle that had its vertex at the center of the circle and spanned exactly 7 sectors. What is the radian
measure of the angle she graphed?
2.
Jim is winding the minute hand of a clock counterclockwise. He changed the time from 4:30 pm to 2:15 pm. How far
did the tip of the minute hand move?
Evaluating trig and inverse trig functions (4.2, 4.3, 4.4, 4.7):
3.
Find the exact value of each expression or write undefined.
a. sin 210°
g. sec
b. cos
15𝜋
12
h. sin
m. cos −1 (−1)
3𝜋
4
17𝜋
2
d. csc 450°
𝜋
n. sin−1 (−
√3
)
2
s. sin−1 (sin
w. tan(tan−1 5)
x. tan−1 (tan
15𝜋
3
√3
)
3
t. cos(cos−1 0.6)
f. 4 tan
k. cos−1 (−
j. cos ( 2 + 30𝜋)
o. tan−1 (−
11𝜋
)
9
e. cot
𝜋
i. sin ( 4 + 81𝜋)
2𝜋
)
3
r. sin−1 (sin
17𝜋
6
c. tan
17𝜋
𝜋
sin 3
4
√3
)
2
+ cos
7𝜋
𝜋
sec 6
6
l. sin−1
√2
2
𝜋
p. tan−1 (1)
q. sin−1 (sin 3 )
u. cos−1 (cos
4𝜋
)
5
𝜋
v. cos (cos−1 2 )
3𝜋
)
4
4.
Use a right triangle, placed in the appropriate quadrant, to find the exact value of the expression or simplify the
expression.
3
2
a. csc [cos −1 (− 5)]
5.
b. sin [cos−1 (5 𝑥)] ; 𝑥 < 0
Let 𝑓(𝑥) = 4 sin 𝑥, 𝑔(𝑥) = cos(4𝑥), and ℎ(𝑥) = tan−1 𝑥. Find the exact value of each expression.
𝜋
𝜋
2
4
𝜋 𝜋
−
2
4
𝑔( )−𝑔( )
a.
17𝜋
13𝜋
)−𝑔( )+
2
3
b. 𝑓 (
ℎ (−
√3
)
3
Graphing trig functions (4.5, 4.6):
6.
Graph at least one period of each function. Label all important values on the 𝑥 and 𝑦 axis.
a. 𝑦 = 3 cos(2𝜋𝑥 + 𝜋)
1
𝜋
e. 𝑦 = −3 tan (2 𝑥 + 10)
1
𝜋
b. 𝑦 = 4 sin (3𝑥 − 2 )
1
1
f. 𝑦 = 2 csc (3 𝑥)
1
c. 𝑦 = −4sin 3 𝑥 + 1
𝜋
𝜋
g. 𝑦 = −2 sec ( 5 𝑥 + 20)
π
d. 𝑦 = tan 2 𝑥
1
𝜋
h. 𝑦 = − 2 cot ( 4 𝑥)
7.
Find an equation for the cosine function that has two consecutive 𝑥-intercepts at 𝑥 = −1 and 𝑥 = 1, and has a yintercept at 𝑦 = −5.
8.
𝜋
𝜋
Find an equation for the tangent function that has two consecutive 𝑥-intercepts at 𝑥 = − 4 and 𝑥 = 2 , and passes
𝜋
through the point (− 16 , −3).
Applications (4.8, 6.1, 6.2):
9.
Information about a triangle with angles 𝐴, 𝐵, 𝐶 and opposite side lengths 𝑎, 𝑏, 𝑐 is given. Solve each triangle.
Round side lengths to the nearest tenth and angle measures to the nearest degree.
a. 𝐴 = 41°, 𝐶 = 77°, 𝑎 = 10
b. 𝑎 = 2, 𝑏 = 3, 𝑐 = 4
c. 𝐶 = 81°, 𝑐 = 11, 𝑏 = 12
d. 𝐶 = 47, 𝑎 = 131, 𝑐 = 98
10.
Refer to the figure shown. DCA  50, ECB  38, AC  10 , CB  15. Find the measure of side AB to
the
D
C
E
 nearest tenth.





A
B
11.
A 20 ft flagpole casts a 7 ft shadow. Find the angle of elevation of the sun.
12.
The leaning tower of Pisa was originally perpendicular to the ground. Because of sinking into the earth, it now
leans at a certain angle, 𝜃, from its original position. An observer, standing 150 feet from of the tower (on the side the
tower leans toward), notes that the angle of elevation to the top of the tower is 53°. The tower is 179 feet tall.
Approximate the leaning angle, 𝜃.
13.
The length of daylight present each day is a periodic function of the proportion of a full year that has passed. If the
longest day of the year has 14 hours of daylight, and occurs half a year after December 16th, and the shortest day of the
year is December 16th, which only has 10 hours of daylight, express the hours of daylight per day as a function of the
proportion of a year that has passed since December 16th.
(Hint: Your function should take the form: 𝑓(𝑝) = 𝐿 + 𝐴 cos 𝜔𝑝, where 𝑝 is a proportion of the year after Dec. 16th—
by the way, a proportion is just a percentage written as a decimal.)
Verifying identities (5.1, 5.2, 5.3):
14.
Verify each identity.
a. sin 𝑥 + cot 𝑥 cos 𝑥 = csc 𝑥
b.
1+sec 𝜃
sin 𝜃+tan 𝜃
= csc 𝜃
d. (tan 𝛽 + cot 𝛽)(cos 𝛽 + sin 𝛽) = csc 𝛽 + sec 𝛽
e.
c.
sin 𝑢
1+cos 𝑢
1+sin 2𝜃
sin 2𝜃
= csc 𝑢 − cot 𝑢
1
2
= 1 + sec 𝜃 csc 𝜃
15 (con’t)
f.
Verify each identity.
sin(𝑥+2𝑥)
sin 𝑥 cos 𝑥
= 4 cos 𝑥 − sec 𝑥
g. sec(𝛼 + 𝛽) =
sec 𝛼 csc 𝛽
cot 𝛽−tan 𝛼
h. 2 sin2 𝑥 + cos 2𝑥 = 1
Solving trig equations (5.5):
16.
Find the exact solution on the interval [0, 2𝜋)
a. cot 𝑥 + 1 = 0
17.
b. tan2 𝑥 sin 𝑥 = sin 𝑥
c. csc 3 𝑥 − csc 𝑥 = 0
5
d. cos 2 𝑥 = −1
Find ALL solutions to the given equations. Express your answers using exact radian measure.
a. cos 𝑥 = −
√2
2
e. csc 4𝑥 = −2
b. 4 sin2 𝑥 − 3 = 0
f. 2 sin2 𝑥 − cos 𝑥 = 1
c. tan 𝑥 = 2 cos 𝑥 tan 𝑥
d. tan 2𝑥 = 1
g. sin 2𝑥 − cos 𝑥 = 0
Sum, difference, double and half angle formulas (5.2, 5.3):
18.
Use a sum or difference identity to find the exact value of each expression.
a. sin 255°
19.
b. cos
11𝜋
12
2
3
1
3
If 𝑥 and 𝑦 are angles that terminate in Quadrant II, with sin 𝑥 = , and cos 𝑦 = − , find the exact value of the
following:
a. sin(𝑥 + 𝑦)
20.
b. cos 10𝜃
c. sin 20𝜃
Use the half-angle identities to find the exact value of each expression.
a. sin 15 °
22.
c. tan(𝑥 + 𝑦)
Given sin 5𝜃 = −0.8 and cos 5𝜃 = 0.6 find the exact value of each of the following:
a. sin 10𝜃
21.
b. cos(𝑥 + 𝑦)
b. cos
5𝜋
8
Given sec 𝛼 = −4 and 5𝜋 < 𝛼 <
1
2
a. sin 𝛼
1
2
b. cos 𝛼
11𝜋
,
2
find the exact value of each of the following:
Polar coordinates (6.3, 6.4):
23.
Convert the polar equation to rectangular form. In words, describe the graph of the rectangular equation (do not
actually graph).
b. 𝑟 2 sin 2𝜃 = 4
a. 𝑟 = 4 sec 𝜃
24.
c. 2 sin 𝜃 − 3 cos 𝜃 = 𝑟
d. 𝑟 = 5
Convert the rectangular equation to polar form. In each case, solve for 𝑟. In words, describe the graph (do not
actually graph).
b. 𝑥 2 = 8𝑦
a. 𝑦 = 2
25.
c. 𝑦 2 − 𝑥 2 = 4
Sketch the graph of each polar equation.
a. 𝑟 = 2
b. 𝜃 =
𝜋
6
c. 𝑟 = 2 + 4 cos 𝜃
d. 𝑟 = 2 sin 2𝜃
Vectors (6.6):
26.
Sketch each vector as a position vector. Then sketch two more representations of the same vector. Finally, find
the vector’s magnitude.
a. −2𝐢 + 5𝐣
27.
Write the vector (in terms of 𝐢 and 𝐣) whose magnitude ‖𝐯‖ and direction angle 𝜃 are given:
‖𝐯‖ = 6√2,
28.
b. 3𝐣
𝜃 = 135°
Write the vector that begins at point (-2,3) and ends at point (-4,2). Find the magnitude of the vector and use a
calculator to approximate the direction angle.
29.
Determine the unit vector that has the same direction as the vector – 𝐢 + 6𝐣.
30.
Let 𝐯 = −5𝐢 + 2𝐣 and 𝐰 = 𝐢 − 3𝐣 be vectors. Find each of the following vectors.
a. 𝐯 + 𝐰
31.
b. −3𝐰
c. 2𝐯 − 𝐰
Express the following force vector in terms of 𝐢 and 𝐣: A child pulls a sled along level ground by exerting a force of
26 pounds on a handle that makes an angle of 30° with the ground.
Conic Sections (9.1 – 9.3):
32.
Graph each conic section. Be sure to label any vertices and/or asymptotes.
a.
(𝑥−3)2
25
+
(𝑦+1)2
16
=1
b. 𝑥 2 + 4𝑦 2 + 10𝑥 − 8𝑦 + 13 = 0
c. 9𝑦 2 − 𝑥 2 = 1
32 (con’t)
d.
Graph each conic section. Be sure to label any vertices and/or asymptotes.
(𝑥+2)2
25
−
𝑦2
25
=1
e. (𝑦 + 2)2 = 12(𝑥 + 1)
f. 𝑦 2 − 2𝑦 + 8𝑥 + 1 = 0
33.
Find the equation of the ellipse.
34.
Find the equation of the hyperbola that has vertices (0, ±3) and asymptotes 𝑦 = ± 5 𝑥.
3