Graph each number in the complex plane and find its absolute value.
1.z = 4 + 4i
SOLUTION:For z = 4 + 4i, (a, b) = (4, 4). Graph the point (4, 4)in the complex plane.
Use the absolute value of a complex number formula.
2.z = 3 + i
SOLUTION:For z = 3 + i, (a, b) = (3, 1). Graph the point (3,1) in the complex plane.
Use the absolute value of a complex number formula.
3.z = 4 6i
SOLUTION:For z = 4 6i, (a, b) = (4, 6). Graph the point (4, 6) in the complex plane.
Use the absolute value of a complex number formula.
4.z = 2 5i
SOLUTION:For z = 2 5i, (a, b) = (2, 5). Graph the point (2, 5) in the complex plane.
Use the absolute value of a complex number formula.
5.z = 3 + 4i
SOLUTION:For z = 3 + 4i, (a, b) = (3, 4). Graph the point (3, 4)in the complex plane.
Use the absolute value of a complex number formula.
6.z = 7 + 5i
SOLUTION:For z = 7 + 5i, (a, b) = (7, 5). Graph the point (7, 5) in the complex plane.
Use the absolute value of a complex number formula.
7.z = 3 7i
SOLUTION:For z = 3 7i, (a, b) = (3, 7). Graph the point (3, 7) in the complex plane.
Use the absolute value of a complex number formula.
8.z = 8 2i
SOLUTION:For z = 8 2i, (a, b) = (8, 2). Graph the point (8, 2) in the complex plane.
Use the absolute value of a complex number formula.
9.VECTORS The force on an object is given by z = 10 + 15i, where the components are measured in newtons (N). a. Represent z as a vector in the complex plane. b. Find the magnitude and direction angle of the vector.
SOLUTION:a. For z = 10 + 15i, (a, b) = (10, 15). Graph the point (10, 15) in the complex plane. Then draw a vector with an initial point at the origin and a terminal point at (10, 15).
b. Use the absolute value of a complex number formula.
Find the measure of the angle thatthevectormakes with the positive real axis.
The magnitude of the force is about 18.03 newtons atanangleofabout56.31.
Express each complex number in polar form.10.4 + 4i
SOLUTION:4 + 4i
Find the modulus r and argument .
The polar form of 4+ 4i is .
11.2 + i
SOLUTION:2 + i Find the modulus r and argument .
The polar form of 2 + i is (cos 2.68 + i sin 2.68).
12.4 i
SOLUTION:
4 i
Find the modulus r and argument .
The polar form of 4 i is
3 (cos 0.34 + i sin 0.34).
13.2 2i
SOLUTION:2 2i Find the modulus r and argument .
The polar form of 2 2i is
.
14.4 + 5i
SOLUTION:4 + 5i
Find the modulus r and argument .
The polar form of 4+ 5i is (cos0.90+i sin 0.90).
15.2 + 4i
SOLUTION:2 + 4i Find the modulus r and argument .
The polar form of 2 + 4i is 2 (cos 2.03 + i sin 2.03).
16.1 i
SOLUTION:
1 i
Find the modulus r and argument .
The polar form of 1 i is
.
17.3 + 3i
SOLUTION:3 + 3i
Find the modulus r and argument .
The polar form of 3+ 3i is .
Graph each complex number on a polar grid. Then express it in rectangular form.
18.10(cos 6 + i sin 6)
SOLUTION:
The value of r is 10, and the value of is6.Plotthe polar coordinates (10, 6).
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of is
.
19.2(cos 3 + i sin 3)
SOLUTION:
The value of r is 2, and the value of is3.Plotthepolar coordinates (2, 3). Notice that 3 radians is
slightly greater than butlessthan.
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of 2(cos 3 + i sin 3) is 1.98 + 0.28i.
20.
SOLUTION:
The value of r is 4, and the value of is . Plot
the polar coordinates .
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of is
.
21.
SOLUTION:
The value of r is 3, and the value of is . Plot
the polar coordinates .
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of is
i.
22.
SOLUTION:
The value of r is 1, and the value of is .
Plot the polar coordinates .
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of is
.
23.
SOLUTION:
The value of r is 2, and the value of is . Plot
the polar coordinates .
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of is1
i.
24.3(cos 180 + i sin 180)
SOLUTION:
The value of r is 3, and the value of is180.Plot the polar coordinates (3,180).
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of 3(cos 180 + i sin 180) is3.
25. (cos 360 + i sin 360)
SOLUTION:
The value of r is , and the value of is360.
Plot the polar coordinates
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of (cos 360 + i sin 360) is
.
Find each product or quotient and express it in rectangular form.
26.
SOLUTION:Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is . The
rectangular form is .
27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)
SOLUTION:Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is 10(cos 180 + i sin 180). The rectangular form is 10.
28.
SOLUTION:Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form is . The
rectangular form is .
29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)
SOLUTION:Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is . The
rectangular form is 4.
30.
SOLUTION:Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form of the quotient is
. The rectangular form
of the quotient is .
31.
SOLUTION:Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form of the quotient
is . The rectangular form of the
quotient is .
32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)
SOLUTION:Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is . The
rectangular form is .
33.
SOLUTION:Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form of the quotient is .
The rectangular form of the quotient is .
34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)
SOLUTION:Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is . The
rectangular form is .
35.
SOLUTION:Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form of the product.
The polar form of the quotient is
. The rectangular form of the
quotient is .
Find each power and express it in rectangular form.
36.(2 + 2 i)6
SOLUTION:
First, write 2 + 2 i in polar form.
The polar form of 2 + 2 i is .
Now use De Moivres Theorem to find the sixth power.
Therefore, .
37.(12i 5)3
SOLUTION:First, write 12i 5 in polar form.
The polar form of 12i 5 is . Now use De Moivres Theorem to find the third power.
Therefore, .
38.
SOLUTION:
is already in polar form. Use De
Moivres Theorem to find the fourth power.
39.( i)3
SOLUTION:
First, write i in polar form.
The polar form of i is . Now use De Moivres Theorem to find the third power.
Therefore, .
40.(3 5i)4
SOLUTION:First, write 3 5i in polar form.
The polar form of 3 5i is
. Now use De
Moivres Theorem to find the fourth power.
Therefore, .
41.(2 + 4i)4
SOLUTION:First, write 2 + 4i in polar form.
The polar form of 2 + 4iis
. Now use De
Moivres Theorem to find the fourth power.
Therefore, .
42.(3 6i)4
SOLUTION:First, write 3 6i in polar form.
The polar form of 3 6i
is . Now use De
Moivres Theorem to find the fourth power.
Therefore, .
43.(2 + 3i)2
SOLUTION:First, write 2 + 3i in polar form.
The polar form of 2 + 3i
is . Now use De
Moivres Theorem to find the second power.
Therefore, .
44.
SOLUTION:
is already in polar form. Use De
Moivers Theorem to find the third power.
45.
SOLUTION:
is already in polar form. Use De
Moivers Theorem to find the fourth power.
46.DESIGN Stella works for an advertising agency. She wants to incorporate a design comprised of regular hexagons as the artwork for one of her proposals. Stella can locate the vertices of one of the central regular hexagons by graphing the
solutions to x6 1 = 0 in the complex plane. Find
the vertices of this hexagon.
SOLUTION:
The equation x
