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Exam #3 REVIEW - 5.1, 6.1-6.7
Name_________________________________________________________________________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Complete the identity.
1) sec4 x + sec2 x tan2 x - 2 tan4 x = ?
1)
2)
(cot x + 1)(cot x + 1) - csc2 x
=?
cot x
2)
3)
sin2 x - cos2 x
=?
1 - cot2 x
3)
Solve the triangle.
4)
4)
80°
7
55°
Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle,
two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle measures
to the nearest degree.
5) B = 39°, b = 4, a = 22
5)
Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
6)
7
6)
6
8
7) a = 6, b = 8, C = 106°
7)
1
Match the point in polar coordinates with either A, B, C, or D on the graph.
8) -2, -
8)
2
Find another representation, (r, ), for the point under the given conditions.
9) 2,
10) 8,
2
6
, r < 0 and 0 <
<2
, r > 0 and -2 <
9)
<0
10)
Polar coordinates of a point are given. Find the rectangular coordinates of the point.
3
11) 9,
4
The rectangular coordinates of a point are given. Find polar coordinates of the point. Express
12) (7, -7)
13) (5 3, 5)
11)
in radians.
12)
13)
Polar coordinates of a point are given. Find the rectangular coordinates of the point.
14) (-7, 120°)
14)
Convert the rectangular equation to a polar equation that expresses r in terms of .
15) 8x - 7y + 10 = 0
15)
16) (x - 13)2 + y2 = 169
16)
Convert the polar equation to a rectangular equation.
17) r = -5 cos
17)
18) r2 sin 2 = 9
18)
2
Test the equation for symmetry with respect to the given axis, line, or pole.
19) r = 4 sin ; the pole
20) r = 6 + 2 sin ; the line
=
19)
20)
2
Graph the polar equation.
21) r = 3 + sin
21)
22) r = 3 sin 2
22)
23) r2 = 4 cos (2 )
23)
3
Plot the complex number.
24) -1 - 2i
24)
Find the absolute value of the complex number.
25) z = -15 + 4i
25)
Write the complex number in polar form. Express the argument in degrees.
26) 3 - 4i
26)
Write the complex number in polar form. Express the argument in radians.
27) 2 - 2i
27)
Write the complex number in rectangular form.
28) -5(cos 120° + i sin 120°)
28)
29) -5(cos
3
3
)
+ i sin
4
4
29)
Find the product of the complex numbers. Leave answer in polar form.
30) z 1 = 5(cos 20° + i sin 20°)
30)
z 2 = 4(cos 10° + i sin 10°)
3
3
+ i sin
31) z 1 = 6 cos
2
2
z 2 = 12 cos
Find the quotient
z1
z2
31)
5
5
+ i sin
6
6
of the complex numbers. Leave answer in polar form.
32) z 1 = 8 cos + i sin
2
2
z 2 = 3 cos
6
+ i sin
32)
6
4
Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.
33) 2(cos 15° + i sin 15°) 4
33)
Find the quotient
z1
z2
of the complex numbers. Leave answer in polar form.
34) z 1 = 30(cos 40° + i sin 40°)
z 2 = 5(cos 7° + i sin 7°)
34)
Solve the equation in the complex number system.
35) x3 = -64i
35)
36) x3 - 27i = 0
36)
Solve the problem.
37) Let vector u have initial point P1 = (0, 2) and terminal point P2 = (3, 0). Let vector v have
37)
initial point Q1 = (3, 0) and terminal point Q2 = (6, -2). u and v have the same direction.
Find u and v . Is u = v?
Sketch the vector as a position vector and find its magnitude.
38) v = 3i - 4j
38)
Let v be the vector from initial point P1 to terminal point P2 . Write v in terms of i and j.
39) P1 = (6, 4); P2 = (-5, -4)
39)
Find the specified vector or scalar.
40) u = -7i - 3j, v = -5i + 7j; Find u + v.
40)
41) v = -7i + 2j; Find 9v .
41)
42) u = -9i - 2j, v = 5i + 7j; Find u - v.
42)
Write the vector v in terms of i and j whose magnitude v and direction angle
43) v = 10, = 120°
5
are given.
43)
Use the given vectors to find the specified scalar.
44) u = 4i - 8j and v = 14i - 13j; Find u · v.
44)
45) v = 6i + 2j; Find v · v.
45)
46) u = -6i + 10j, v = -7i - 6j; Find (-5u) · v.
46)
Find the angle between the given vectors. Round to the nearest tenth of a degree.
47) u = -3i + 6j, v = 5i + 2j
48) u = -i + 4j, v = 2i - 5j
47)
48)
Use the dot product to determine whether the vectors are parallel, orthogonal (perpendicular), or neither.
49) v = 2i + j, w = i - 2j
49)
Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.
50) v = 3i - j, w = 6i - 2j
6
50)
Answer Key
Testname: EXAM_3_REVIEW
1) 3 sec4 x - 2
2) 2
3) sin2 x
4)
5)
6)
7)
8)
B = 45°, a = 8.11, c = 9.75
no triangle
A = 58°, B = 47°, C = 75°
c = 11.2, A = 31°, B = 43°
A
3
9) -2,
2
10) 8, 11)
11
6
-9 2 9 2
,
2
2
12) 7 2,
13) 10,
14)
7
4
6
7 -7 3
,
2
2
15) r =
(8 cos
-10
- 7 sin )
16) r = 26 cos
5 2
25
+ y2 =
17) x +
2
4
18) xy =
9
2
19) may or may not have symmetry about the pole
20) has symmetry with respect to the line
=
2
21)
7
Answer Key
Testname: EXAM_3_REVIEW
22)
23)
24)
25) 241
26) 5(cos 306.9° + i sin 306.9°)
7
7
+ i sin
27) 2 2 cos
4
4
28)
5 -5 3
i
+
2
2
29)
5 2 -5 2
i
+
2
2
30) 20(cos 30° + i sin 30°)
8
Answer Key
Testname: EXAM_3_REVIEW
31) 72 cos
3
+ i sin
3
32)
8
cos + i sin
3
3
3
33)
34)
35)
36)
37)
38)
8 + 8 3i
6(cos 33° + i sin 33°)
4(cos 90° + i sin 90°), 4(cos 210° + i sin 210°), 4(cos 330° + i sin 330°)
3(cos 30° + i sin 30°), 3(cos 150° + i sin 150°), 3(cos 270° + i sin 270°)
u = 13, v = 13; yes
v =5
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
v = -11i - 8j
-12i + 4j
9 53
-14i - 9j
v = -5i + 5 3j
160
40
90
94.8°
172.2°
orthogonal
parallel
9