Math 102: Trigonometry
Test 4 Winter 2016
Dr. Matsumoto
Name___________________________________
Each multiple-choice question is worth 3 points without partial credit. Other questions are worth 5 points each unless
otherwise specified.You MAY NOT use a calculator on this test.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the magnitude and direction angle for each vector. Give the angle in [0,360°).
1) -3 3, 3
A) 6; 150°
B) 12; 150°
C) 6; 330°
D) 12; 30°
Find the product. Write the answer in standard form.
2) (4 + i)(-4 - i)
A) -17
B) -15 - 8i
C) -15
D) -17 - 8i
B) 2i
C) 2
D) -2
B) -1
C) -i
D) 1
Find the quotient. Write the answer in standard form.
2
3)
-i
A) -2i
Simplify the power of i.
4) i49
A) i
1
Write the complex number in rectangular form.
π
π
5) 3 cos + i sin
3
3
A)
3+i
B)
3
3
i
+
6
6
C)
3 3 3
i
2
2
D)
3 3 3
i
+
2
2
Write the complex number in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°).
6) 9 + 9i
A) 9 2(cos 135° + i sin 135°)
B) 9 2(cos 225° + i sin 225°)
C) 9 2(cos 315° + i sin 315°)
D) 9 2(cos 45° + i sin 45°)
The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°.
7) (0, -2)
A) (2, 0°)
B) (2, 270°)
C) (2, 90°)
D) (2, 180°)
A point is given in polar coordinates. Give the rectangular coordinates for the point.
8) (6, 225°)
A) (-3π, -3π)
B) (-3 3, -3 3)
C) (3 2, 3 2)
2
D) (-3 2, -3 2)
For the given rectangular equation, give its equivalent polar equation.
9) x2 + y2 = 64
A) r = 8 cos θ
B) r = 64
Find an equivalent equation in rectangular coordinates.
10) r = cos θ
A) x2 + y2 = y
B) (x + y)2 = x
C) r = 8
D) r = 8 sin θ
C) x2 + y2 = x
D) (x + y)2 = y
Answer each question.
Find the quotient. Write the answer in standard form.
1+i
11)
4-i
Use the dot product to determine whether the angle between the two vectors is acute, right, or obtuse. Be sure to explain
your reasoning.
12) 4, -1 , 8, 32
3
Find an expression that represents the the angle between the two vectors.
13) 4, -3 , 5, 5
Write an expression for the length of b (the side AC; opposite of angle B).
14)
30 m
Find an expression that represents the measure of angle A (angle opposite to side a) in the following triangle.
15) a = 25 ft
b = 30 ft
c = 40 ft
4
Find the product. and simplify. Write the product in rectangular (standard) form, using exact values.
16) [8(cos 300° + i sin 300°)] [6(cos 330° + i sin 330°)]
Find the following quotient, and write the quotient in rectangular form, using exact values.
25(cos 240° + i sin 240°)
17)
5(cos 60° + i sin 60°)
Find the given power using DeMoivre's Theorem. Write answer in rectangular form. Simplify as much as possible.
18) cos 30° + i sin 30° 12
19) (1 - i)4
5
Find all cube roots of the complex number.
20) -8i
Find all specified roots. Leave your answers in trigonometric form using radians.
21) Fifth roots of 1.
Find all solutions of the equation.
22) x4 + 16 = 0
6
Graph the polar equation for θ in [0°, 360°).
23) r = 4 + 4 cos θ
r
Graph the curve defined by the following parametric equations. Find a rectangular equation for the curve.
24) x = 4 cos t, y = 7 sin t, for t in [0, 2π]
y
10
-10
10
x
-10
7