9-1 study guide and intervention - mrs. fruge · 9-2 study guide and intervention graphs of polar...

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 9 5 Glencoe Precalculus

9-1 Study Guide and Intervention Polar Coordinates

Graph Polar Coordinates A polar coordinate system uses distances and angles to record the position of a point.

The location of a point P can be identified by polar coordinates of the form (r, θ), where r is the directed distance from the pole, or origin, to point P and θ is the measure of the directed angle formed by the ray from the pole to point P and

the polar axis.

Example: Graph each point.

a. P (3,𝜋

4)

Sketch the terminal side of an angle measuring 𝜋

4 radians in standard position.

Since r is positive (r = 3), find the point on the terminal side of the angle that is

3 units from the pole. Notice point P is on the third circle from the pole.

b. Q(–2.5, –120°)

Negative angles are measured clockwise. Sketch the terminal side of an angle of

–120° in standard position.

Since r is negative, extend the terminal side of the angle in the opposite direction.

Find the point Q that is 2.5 units from the pole along this extended ray.

Exercises

Graph each point on a polar grid.

1. R(3, 60°) 2. Q (4, – 4𝜋

3) 3. A(–2.5, –150°)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-1 Study Guide and Intervention (continued)

Polar Coordinates

Graphs of Polar Equations An equation expressed in terms of polar coordinates is called a polar equation. A polar

graph is the set of all points with coordinates (r, θ) that satisfy a given polar equation. The graphs of polar equations like r = k and θ = k, where k is a constant, are considered basic in the polar coordinate system. The solutions of r = k are

ordered pairs of the form (k, θ) where θ is any real number. The solutions of θ = k are ordered pairs of the form (r, θ)

where r is any real number.

Example: Graph each polar equation.

a. r = 3

The solutions of r = 3 are ordered pairs of the form (3, θ), where θ is any real number.

The graph consists of all the points that are 3 units from the pole, so the graph is

a circle centered at the origin with radius 3.

b. θ = –60°

The solutions of θ = –60° are ordered pairs of the form (r, –60°), where r is any

real number. The graph consists of all the points on the line that make an angle of –60° with the positive polar axis.

Exercises

Graph each polar equation.

1. r = 4 2. θ = 3𝜋

4 3. θ = –300°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-1 Practice Polar Coordinates

Graph each point on a polar grid.

1. (2.5, 0°) 2. (−2,𝜋

4) 3. (–1, –30°)

Graph each polar equation.

4. r = 2 5. θ = 60° 6. r = 4

7. LANDSCAPING A landscape architect has created a blueprint for the landscape design at a new building being

constructed at a retirement community.

a. The architect has placed a gazebo at (3, –135°). Graph this point.

b. The design calls for a bench at (–4, 85°) and a pond at (1, 105°). Find the distance in feet between the pond and the bench.

4.95 units

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-2 Study Guide and Intervention Graphs of Polar Equations

Graphs of Polar Equations A polar graph is the set of all points with coordinates (r, θ) that satisfy a given polar

equation. The position and shape of polar graphs can be altered by multiplying or adding to either the function or θ.

Example 1: Graph the polar equation r = 2 cos 2θ.

Make a table of values on the interval [0, 2π].

θ 0 𝜋

6

𝜋

4

𝜋

3

𝜋

2

2𝜋

3

3𝜋

4

5𝜋

6 π

7𝜋

6

5𝜋

4

4𝜋

3

3𝜋

2

5𝜋

3

7𝜋

4

11𝜋

6 2π

r = 2 cos 2θ 2 1 0 1 –2 –1 0 1 2 1 0 –1 –2 –1 0 1 2

Graph the ordered pairs (r, θ) and connect with a smooth curve.

This type of curve is called a rose. Notice that the farthest points are 2 units from the pole and the rose has 4 petals.

Example 2: Graph the polar equation r = 1 + 2 sin θ. Round each r-value to the nearest tenth.

θ 0 𝜋

6

𝜋

4

𝜋

3

𝜋

2

2𝜋

3

3𝜋

4

5𝜋

6 π

7𝜋

6

5𝜋

4

4𝜋

3

3𝜋

2

5𝜋

3

7𝜋

4

11𝜋

6 2π

r = 1 + 2 sin θ 1 2 2.4 2.7 3 2.7 2.4 2 1 0 –0.4 –0.7 –1 –0.7 –0.4 0 1

Graph the ordered pairs and connect them with a smooth curve. This type of curve

is called a limaçon.

Exercises

Graph each equation by plotting points.

1. r = 2 sin θ 2. r = 2 + 2 sin θ 3. r = 1 – 3 cos θ

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Graphs of Polar Equations

Classic Polar Curves The graph of a polar equation is symmetric with respect to the polar axis if it is a function of

cos θ, and to the line θ = 𝜋

2 if it is a function of sin θ. It is symmetric to the pole if replacing (r, θ) with (–r, θ) or (r, π + θ)

produces an equivalent equation. Knowing whether a graph is symmetric can reduce the number of points needed to

sketch it.

Example: Determine the symmetry, zeros, and maximum r-values of r = 𝟏

𝟐 sin 2θ. Then use this information to

graph the function.

The function is symmetric with respect to the line θ = 𝜋

2, so you can find points on

the interval [− 𝜋

2,

𝜋

2] and then use line symmetry to complete the graph. To find the

zeros and the maximum r-value, sketch the graph of the rectangular function

y = 1

2 sin 2x.

From the graph, you can see that |y| = 1

2 when x =

𝜋

4, and

3𝜋

4 and y = 0 when x = 0,

𝜋

2, and π. That means that |r| has a

maximum value of 1

2 when θ =

𝜋

4 or

3𝜋

4 and r = 0 when θ = 0,

𝜋

2, or π. Use these and a few additional points to sketch the

graph of the function.

Use the axis of symmetry to complete the graph after plotting points on [− 𝜋

2,

𝜋

2].

Exercises

Use symmetry, zeros, and maximum r-values to graph each function.

1. r = 4 sin 3θ 2. r = 3 cos 2θ

Symmetry line θ = 𝝅

𝟐, |r | = 4 at

𝝅

𝟔,

𝝅

𝟐,

𝟓𝝅

𝟔,

𝟕𝝅

𝟔, Symmetric to polar axis, θ =

𝝅

𝟐, and origin; |r | = 3

𝟑𝝅

𝟐; r = 0 when θ = 0,

𝝅

𝟑,

𝟐𝝅

𝟑, 𝝅,

𝟒𝝅

𝟑, and

𝟓𝝅

𝟑. at 0,

𝝅

𝟐, 𝝅,

𝟑𝝅

𝟐; r = 0 when θ =

𝝅

𝟒,

𝟑𝝅

𝟒,

𝟓𝝅

𝟒, and

𝟕𝝅

𝟒.

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-2 Practice Graphs of Polar Equations

Use symmetry to graph each equation.

1. r = 3 + 2 sin θ 2. r = 2 + 2 cos θ

Identify the type of curve given by each equation. Then use symmetry, zeros, and maximum r-values to

graph the function.

3. r = 1 + cos θ 4. r = 3 sin 3θ 5. r = 2 cos θ

6. DESIGN Mikaela is designing a border for her stationery. Suppose she uses a rose curve.

Determine an equation for designing a rose that has 8 petals with each petal 4 units long.

cardioid rose circle

symmetric to polar symmetric to θ = 𝝅

𝟐, symmetric to polar

axis, zero 𝝅, |r | = 2 zeros 0, 𝝅

𝟑, 𝟐𝝅

𝟑, 𝝅,

𝟒𝝅

𝟑, axis, zeros

𝝅

𝟐, 𝟑𝝅

𝟐,

at 0, 2𝝅 𝟓𝝅

𝟑, |r | = 3 at

𝝅

𝟔, 𝟓𝝅

𝟔, 𝟑𝝅

𝟐 |r | = 2 at 0, 𝝅

Sample answer: r = 4 sin 4θ

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-3 Study Guide and Intervention Polar and Rectangular Forms of Equations

Polar and Rectangular Coordinates If a point P has polar coordinates (r, θ), then the rectangular coordinates

(x, y) of P are given by x = r cos θ and y = r sin θ. If a point P has rectangular coordinates (x, y), then the polar

coordinates (r, θ) of P are given by r = √𝑥2 + 𝑦2 and θ = tan−1 𝑦

𝑥, when x > 0, and θ = tan−1

𝑦

𝑥 + π, when x < 0.

Example 1: Find rectangular coordinates for point P with the polar coordinates (𝟑,𝟑𝝅

𝟒).

For P(3,3𝜋

4), r = 3 and θ =

3𝜋

4. Use the conversion formulas.

x = r cos θ y = r sin θ

= 3 cos 3𝜋

4 = 3 sin

3𝜋

4

= 3 (−√2

2) or −

3√2

2 = 3(

√2

2) or

3√2

2

The rectangular coordinates of P are (−3√2

2,

3√2

2 ), or (–2.12, 2.12) to the nearest hundredth.

Example 2: Find two pairs of polar coordinates for point R with the rectangular coordinates (5, –9).

For R(5, –9), x = 5 and y = –9.

r = √𝑥2 + 𝑦2 θ = tan−1 𝑦

𝑥

= √52 + (−9)2 = tan−1 −9

5

= √106 or about 10.30 = –1.06

One pair of polar coordinates for R is (10.30, –1.06). To obtain a second pair of polar coordinates for R, you can add 2π to the θ-value. This results in (10.30, –1.06 + 2π) or (10.30, 5.22).

Exercises

Find rectangular coordinates for each point with the given polar coordinates.

1. (20, –60°) 2. (−1,5𝜋

6) 3. (6, –30°) 4. (3,

𝜋

3)

Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 ≤ θ ≤ 2π.

5. (2, –2) 6. (3, 5)

(10, –10√𝟑) (√𝟑

𝟐, −

𝟏

𝟐) (3√𝟑, –3) (

𝟑

𝟐,

𝟑√𝟑

𝟐)

(𝟐√𝟐, −𝝅

𝟒) and (𝟐√𝟐,

𝟕𝝅

𝟒) (5.8, 1.0) and (–5.8, 4.2)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Polar and Rectangular Forms of Equations

Polar and Rectangular Equations You can also use the relationships 𝑟2 = 𝑥2 + 𝑦2, x = r cos θ and y = r sin θ,

and tan θ = 𝑦

𝑥 to convert between rectangular equations and polar equations.

Example 1: Identify the graph of the rectangular equation y = –3𝒙𝟐. Then write the equation

in polar form. Support your answer by graphing the polar form of the equation.

The graph of y = –3𝑥2 is a parabola with vertex at the origin that opens down.

y = –3𝑥2 Original equation

r sin θ = –3(𝑟 cos 𝜃)2 x = r cos θ and y = r sin θ

r sin θ = –3𝑟2 cos2 θ Multiply.

sin θ

−3 cos2 θ = r Divide each side by –3r cos2 θ.

– 1

3 tan θ sec θ = r Quotient and Reciprocal Identities

The graph of the polar equation r = – 1

3 tan θ sec θ is a parabola with vertex at the pole that opens down.

Example 2: Write the polar equation r = 5 cos θ in rectangular form and then identify its graph.

Support your answer by graphing the polar form of the equation.

r = 5 cos θ Original equation

𝑟2 = 5r cos θ Multiply each side by r.

𝑥2 + 𝑦2 = 5x 𝑟2 = 𝑥2 + 𝑦2 and r cos θ = x

𝑥2 – 5x + 𝑦2 = 0 Subtract 5x from each side.

Because in standard form this equation is (𝑥 – 2.5)2 + 𝑦2 = 6.25, you can

identify the graph of this equation as a circle centered at (2.5, 0) with radius 2.5, as supported by the graph of r = 5 cos θ.

Exercises

Identify the graph of each rectangular equation. Then write the equation in polar form.


1. x = 5 2. y = –x

vertical line at x = 5; r = 5 sec θ θ = 𝟑𝝅

𝟒, line through (0, 0) , (–1, 1)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-3 Practice Polar and Rectangular Forms of Equations

Find the rectangular coordinates for each point with the given polar coordinates.

1. (6, 120˚) 2. (–4, 45˚) 3. (4,𝜋

6)

Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 ≤ θ < 2π.

4. (2, 2) 5. (2, –3) 6. (−3, √3)

Identify the graph of each rectangular equation. Then write the equation in polar form.


7. 𝑥2 + 𝑦2 = 9 8. y = 3

Write each equation in rectangular form and then identify its graph. Support your answer by graphing the polar

form of the equation.

9. r = 4 10. r cos θ = 5

11. SURVEYING A surveyor records the polar coordinates of the location of a landmark as (40, 62°).

What are the rectangular coordinates?

(–3, 3√𝟑) (–2√𝟐, –2√𝟐) (2√𝟑, 2)

(𝟐√𝟐 ,𝝅

𝟒) , (−𝟐√𝟐,

𝟓𝝅

𝟒) (3.61, 5.30), (–3.61, 2.16) (𝟐√𝟑,

𝟓𝝅

𝟔), (−𝟐√𝟑,

𝟏𝟏𝝅

𝟔)

r = ±3 ; circle with center r sin θ = 3 or r = 3 csc θ; (0, 0), radius 3 horizontal line at y = 3

x2 + y2 = 16; circle with x = 5; vertical line through center (0, 0), radius 4 (5, 0)

(18.78, 35.32)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-4 Study Guide and Intervention Polar Forms of Conic Sections

Identify Polar Equations of Conics

The conic section with eccentricity e > 0, d > 0, and focus at the pole has the polar equation:

• r = 𝑒𝑑

1 + 𝑒 cos 𝜃 if the directrix is the vertical line x = d.

• r = 𝑒𝑑

1 − 𝑒 cos 𝜃 if the directrix is the vertical line x = –d.

• r = 𝑒𝑑

1 + 𝑒 sin 𝜃 if the directrix is the horizontal line y = d.

• r = 𝑒𝑑

1 − 𝑒 sin 𝜃 if the directrix is the horizontal line y = –d.

Eccentricities of conics:

Ellipse: 0 < e < 1

Parabola: e = 1

Hyperbola: e > 1

Example: Determine the eccentricity, type of conic, and equation of the directrix for the

polar equation r = 𝟔

𝟑 + 𝟑 𝐬𝐢𝐧 𝜽.

Write the equation in standard form r = 𝑒𝑑

1 + 𝑒 sin 𝜃.

r = 6

3 + 3 sin 𝜃 Original equation

= 3(2)

3(1 + sin 𝜃) Factor the numerator and denominator.

= 2

1 + sin 𝜃 Divide the numerator and denominator by 3.

The denominator tells us that e = 1; therefore, the conic is a parabola. For polar equations of this form, the equation

of the directrix is y = d. From the numerator, we know that ed = 2, so d = 2 ÷ 1 or 2. Therefore, the equation of the directrix is y = 2.

Exercises

Determine the eccentricity, type of conic, and equation of the directrix for each polar equation.

1. r = 2

1 − 1

2 sin 𝜃

2. r = 3

1 + 3 sin 𝜃 3. r =

4

2 − 3 sin 𝜃

4. r = 3

4 − 2 cos 𝜃 5. r =

8

4 + 3 sin 𝜃 6. r =

9

3 − 6 cos 𝜃

e = 𝟏

𝟐; ellipse; e = 3; hyperbola; e =

𝟑

𝟐; hyperbola;

y = –4 y = 1 y = −𝟒

𝟑

e = 𝟏

𝟐; ellipse; e =

𝟑

𝟒; ellipse; e = 2; hyperbola

x = −𝟑

𝟐 y =

𝟖

𝟑 x = −

𝟑

𝟐

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Polar Forms of Conic Sections

Polar Equations of Conics You can use the characteristics of a conic to write and graph a polar equation of the conic.

Example: Write and graph a polar equation and directrix for the conic with e = 𝟐

𝟑 and directrix x = 5.

Because e = 2

3, the conic is an ellipse.

The directrix x = 5 is to the right of the pole, so the equation is of the form

r = 𝑒𝑑

1 + 𝑒 cos 𝜃.

Use the values for e and d to write the equation.

r = 𝑒𝑑

1 + 𝑒 cos 𝜃 Polar form of conic with directrix x = d

r = (

2

3)(5)

1 + 2

3 cos 𝜃

e = 2

3 and d = 5

r = 10

3 + 2 cos 𝜃 Simplify and multiply by

3

3.

Sketch the graph of this polar equation and its directrix. The graph is an ellipse with its directrix to the right of the pole.

Exercises

Write and graph a polar equation and directrix for the conic with the given characteristics.

1. e = 1; directrix: x = 1.5 2. e = 3; directrix: y = -3 3. e = 1

3; directrix: x =

3

2

r = 𝟑

𝟐 + 𝟐 𝐜𝐨𝐬 𝜽 r =

𝟗

𝟏− 𝟑 𝐬𝐢𝐧 𝜽 r =

𝟑

𝟔 + 𝟐 𝐜𝐨𝐬 𝜽

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-4 Practice Polar Forms of Conic Sections

Determine the eccentricity, type of conic, and equation of the directrix for each polar equation.

1. r = 12

2 − 6 cos 𝜃 2. r =

15

4 + 3 sin 𝜃

Write and graph a polar equation and directrix for the conic with the given characteristics.

3. e = 3; vertices at (–6, 0) and (–3, 0) 4. e = 0.4; directrix: y = 4

Write each polar equation in rectangular form.

5. r = 1

3 − 3 cos 𝜃 6. r =

42

5 − 2 cos 𝜃

Determine the type of conic for each polar equation. Then graph each equation.

7. r = 3

4 − 2 sin (𝜃 + 𝜋

3) 8. r =

5

1 + cos (𝜃 + 𝜋

3)

9. LANDSCAPING A landscaper is designing a garden, and hedges will divide two parts of the garden.

The polar equation r = 6

1 + 2 sin 𝜃 describes the pattern of the hedges. Determine the conic for this equation.

3; hyperbola; x = –2 0.75; ellipse; y = 5

r = 𝟏𝟐

𝟏 –𝟑 𝐜𝐨𝐬 𝜽 r =

𝟖

𝟓 +𝟐 𝐬𝐢𝐧 𝜽

9y2 = 6x + 1 (𝒙 − 𝟒)𝟐

𝟏𝟎𝟎 +

𝒚𝟐

𝟖𝟒 = 1

ellipse parabola

hyperbola

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-5 Practice Complex Numbers and De Moivre’s Theorem

Graph each number in the complex plane and find its absolute value.

1. 2 + 3i 2. –1 + 4i

Express each complex number in polar form.

3. 2 + 2√3i 4. 3√3 – 3i

Graph each complex number on a polar grid. Then express it in rectangular form.

5. 4(cos 3π

4 + 𝒊 sin

3π

4) 6. 5(cos 150° + i sin 150°)

Find each product or quotient and express it in rectangular form.

7. 2(cos π

6 + 𝒊 sin

π

6) ⋅ 5(cos

π

3 + 𝒊 sin

π

3)

8. 8(cos 240° + i sin 240°) ÷ 4(cos 210° + i sin 210°)

Find each power and express it in rectangular form.

9. ( −√3 + 𝒊)5 10. (1 + 𝒊)10

Find all the distinct pth roots of the complex number.

11. fourth roots of –8 + 8√3i 12. seventh roots of i

13. ELECTRICITY Find the current in a circuit with a voltage of 12 volts and an impedance of 2 – 4 j ohms. Use the

formula, E = I ⋅ Z, where E is the voltage measured in volts, I is the current measured in amperes, and Z is the

impedance measured in ohms.

(Hint: Electrical engineers use j as the imaginary unit, so they write complex numbers in the form a + bj. Express each number in polar form, substitute values into the formula, and then express the current in rectangular form.)

3.61 4.12

4(𝐜𝐨𝐬 𝝅

𝟑 + 𝒊 𝐬𝐢𝐧

𝝅

𝟑) 6(𝐜𝐨𝐬

𝟏𝟏𝝅

𝟔 + 𝒊 𝐬𝐢𝐧

𝟏𝟏𝝅

𝟔)

–2√𝟐 + 2√𝟐𝒊 −𝟓√𝟑

𝟐+ (

𝟓

𝟐) 𝒊

10(𝐜𝐨𝐬 𝝅

𝟐 + 𝒊 𝐬𝐢𝐧

𝝅

𝟐); 10i

2(cos 30° + i sin 30°); √𝟑 + i

16√𝟑 + 16i 32i

√𝟑 + i, –1 + √𝟑i, – √𝟑 – i, 1 – √𝟑i • ±0.97 + 0.22i, ± 0.43 + 0.90i, ± 0.78 – 0.62i, –i

1.2 + 2.4j amps

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


9-5 Study Guide and Intervention Complex Numbers and De Moivre’s Theorem

Polar Forms of Complex Numbers You can convert the complex number z = a + bi to polar or trigonometric form

using the formula z = r(cos θ + i sin θ), where r = |z| = √𝑎2 + 𝑏2 and θ = tan−1 𝑏

𝑎 for a > 0, θ = tan−1

𝑏

𝑎 + π for a < 0.

You can also convert a complex number in polar form to rectangular form by evaluating it for given values of r and θ using the formulas a = r cos θ and b = r sin θ.

Example 1: Express 2√𝟑 – 2i in polar form.

First find the modulus. Then find the argument.

r = √𝑎2 + 𝑏2 θ = tan−1 𝑏

𝑎

= √(2√3)2 + (−2)2 = tan−1 2

2√3 or

11𝜋

6

= 4

The polar form of 2√3 – 2i is 4(cos 11𝜋

6 + 𝒊 sin

11𝜋

6) or about 4(cos 5.76 + i sin 5.76).

Example 2: Express z = 2(𝐜𝐨𝐬 𝝅

𝟒 + 𝒊 𝐬𝐢𝐧

𝝅

𝟒) in rectangular form.

Evaluate the trigonometric values and simplify.

2(cos 𝜋

4 + 𝑖 sin

𝜋

4) = 2(

√2

2 + 𝒊 (

√2

2)) or √2 + i√2

The rectangular form of z is √2 + i√2.

Exercises

Express each complex number in polar form.

1. 1 – i 2. 3 + 2i 3. –1 + √3i

Graph each complex number on a polar grid. Then express it in rectangular form.

4. 4(cos 𝜋

6 + 𝒊 sin

𝜋

6) 5. 4√2 (cos

5π

4 + 𝒊 sin

5π

4) 6. 6(cos

5π

3 + 𝒊 sin

5π

3)

√𝟐 (𝐜𝐨𝐬 𝟕𝝅


𝟕𝝅

𝟒) √𝟏𝟑 (cos 33.69° + i sin 33.69°) 2 (𝐜𝐨𝐬

𝟐𝝅

𝟑 + 𝒊 𝐬𝐢𝐧

𝟐𝝅

𝟑)

2√𝟑 + 2i –4 – 4i 3 – 3√𝟑i

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Complex Numbers and De Moivre’s Theorem

Products, Quotients, Powers, and Roots of Complex Numbers Use these formulas to multiply and divide

complex numbers.

Given the complex numbers 𝑧1 = 𝑟1(cos 𝜃1 + i sin 𝜃1) and 𝑧2 = 𝑟2(cos 𝜃2 + i sin 𝜃2),

Product Formula: 𝑧1𝑧2 = 𝑟1𝑟2 [cos (𝜃1 + 𝜃2) + i sin (𝜃1 + 𝜃2)]

Quotient Formula: 𝑧1

𝑧2 =

𝑟1

𝑟2[cos (𝜃1 – 𝜃2) + i sin (𝜃1 – 𝜃2)]

Example: Find 3(𝐜𝐨𝐬 𝛑


𝛑

𝟒) · 4(𝐜𝐨𝐬

𝛑

𝟐 + 𝒊 𝐬𝐢𝐧

𝛑

𝟐), and express it in rectangular form.

3(cos π

4 + 𝑖 sin

π

4) · 4(cos

π

2 + 𝑖 sin

π

2) Original expression

= 3(4)[cos (π

4 +

π

2) + 𝒊 sin (

π

4 +

π

2)] Product Formula

= 12(cos 3π

4 + 𝒊 sin

3π

4) Simplify.

= 12(−√2

2 + 𝒊

√2

2) Evaluate.

= –6√2 + 6i√2 Distributive Property

The polar form of the product is 12(cos 3π

4 + 𝒊 sin

3π

4) and the rectangular form of the product is –6√2 + 6i√2.

You can use De Moivre’s Theorem, [𝑟(cos 𝜃 + 𝒊 sin 𝜃)]𝑛 = 𝑟𝑛(cos nθ + i sin nθ), to find the powers and roots of

complex numbers in polar form.

Exercises

Find each product or quotient and express it in rectangular form.

1. 3(cos π

3 + 𝒊 sin

π

3) · 3(cos

5π

3 + 𝒊 sin

5π

3) 2. 6(cos

π

2 + 𝒊 sin

π

2) ÷ 2(cos

π

3 + 𝒊 sin

π

3)

Find each power and express it in rectangular form.

3. (2 − 2√3𝒊)3 4. (1 − 𝒊)5

9 𝟑√𝟑

𝟐 +

𝟑

𝟐i

–64 –4 + 4i

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