1 TRIG-Fall 2011-Jordan

Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009

Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations

Section 8.1 Complex Numbers

Definitions

i = 1 and is called the “imaginary unit” i2 = -1

aia or a i

Complex Numbers A complex number is any number written in the standard form a + bi, where a is the real part and b is the imaginary part. Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. When working with square roots of negative numbers, convert to i notation before you perform any other operations upon the numbers. Never leave i2 in your answer; convert i2 to –1.

Example 1 Write in standard form: 227

ioriiii 332332233239227

Example 2 Multiply or divide, as indicated, and simplify the answer.

= (-1) = (-1)(3) = -3

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Solving Quadratic Equations Try factoring, extracting roots, completing the square, or the quadratic formula.

If ax2 + bx + c = 0, then

Example 3 Solve: x2 + 48 = 0

x2 = -48 → x = → x = ± i → x = ± 4i Example 4 Solve: 3(3x2 – 2x) = -7 9x2 – 6x + 7 = 0 → Use quadratic formula with a = 9, b = -6, and c = 7

=

Adding/Subtracting Complex Numbers

Add/subtract the real parts and add/subtract the imaginary parts Example 5 Perform the operation and write the result in standard form.

(-3 + i) – (-8 + 2i) = -3 + i + 8 – 2i = -3 + 8 + i – 2i = 5 – i

Multiplying Complex Numbers

Use the distributive property or the FOIL method Example 6 Perform the operation and write the result in standard form. (6 – 2i)(2 – 3i) FOIL: 6(2) – 6(3i) –2i(2) + 2i(3i) 12 – 18i – 4i + 6i2 12 – 22i + 6(-1) 12 – 6 – 22i 6 – 22i

Dividing Complex Numbers Multiply both top and bottom of the fraction by the conjugate of the denominator

Example 7 Perform the operation and write the result in standard form: i

i

2

2

iii

iii

iii

i

i

i

i

5

4

5

3

5

43

)1(4

)1(44

224

224

2

2

2

22

2

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Section 8.2 Trigonometric (Polar) Form of Complex Numbers

The Complex Plane Let the horizontal axis be the real axis and the vertical axis the imaginary axis. Use the real part of the complex number to move left or right and the imaginary part to move up or down. Each complex number a + bi determines a unique vector with initial point (0, 0) and terminal point (a, b). Example 1 Graph each complex number.

a) -2 – 3i b) 1 3 i c) 5 d) 2i imaginary real

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Trigonometric Form of a Complex Number Consider x + yi (x, y) Imaginary

r x y 2 2 tan 1 y

x

x = r cos θ Real y = r sin θ x + yi = r cos θ + (r sin θ)i = r(cos θ + i sin θ) = r cis θ Example 2 Write 2(cos 120˚ + i sin 120˚) in rectangular form. Example 3 Write 4 cis 30˚ in rectangular form. Example 4 Write 3 – 3i in trigonometric form, with θ in the interval [0°, 360°). Example 5 Write -4 + i in trigonometric form, using a calculator as necessary.

x + yi = r(cos θ + i sin θ) = r cis θ

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Section 8.3 The Product and Quotient Theorems

Product Theorem

21212211 cisrrcisrcisr

To multiply two complex numbers, multiply their moduli (r-values) and add their arguments (θ-values). The product theorem can be proven using FOIL and the sum and difference identities for sine and cosine. Example 1 Find the product and write it in rectangular form. [4(cos 50˚ + i sin 50˚)] • [2(cos 10˚ + i sin 10˚)] Example 2 Use a calculator to perform the indicated operation. Give your answer in rectangular form, rounding decimals to four places. (8 cis 88˚)(12 cis 112˚)

Quotient Theorem

To divide two complex numbers, divide their moduli (r-values) and subtract their arguments (θ-values). Example 3 Find the quotient and write it in rectangular form.

16 70 70

4 40 40

(cos sin )

(cos sin )

i

i

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Section 8.4 De Moivre’s Theorem; Powers and Roots of Complex Numbers

Powers of Complex Numbers De Moivre’s Theorem is a procedure for finding powers and roots of complex numbers when the complex numbers are expressed in trigonometric form. Consider the complex number z = r(cos θ + i sin θ). Repeated use of the product theorem from the previous section yields the following pattern: 1. z1 = r(cos θ + i sin θ) = r cis θ 2. z2 = r cis θ · r cis θ = (r · r) cis (θ + θ) = r2 cis 2θ 3. z3 = r cis θ · r2 cis 2θ = (r · r2) cis (θ + 2θ) = r3 cis 3θ This pattern leads to De Moivre’s Theorem. Remember that to use this theorem, the complex number must be expressed in trigonometric form. Example 1 Find the indicated power. Write the answer in rectangular form.

Example 2 Find the indicated power. Write the answer in rectangular form.

De Moivre’s Theorem

If r(cos θ + i sin θ) is a complex number and n is any real number, then

[r(cos θ + i sin θ)]n = rn (cos nθ + i sin nθ).

In compact form, this is written [r cis θ]n = rn cis nθ.

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Roots of Complex Numbers

All n of the nth roots of a complex number have the same magnitude, rn . These roots

are equally spaced along a circle of radius rn and center (0, 0). Therefore, successive

nth roots will have angles that differ by 360

n.

To use this theorem, the complex number must be expressed in trigonometric form. When k exceeds n – 1, the roots will repeat. Example 3 Find the indicated roots. Leave the results in trigonometric form. Graph each root as a vector in the complex plane.

The three cube roots of 2 2 3 i

Example 4 Find all complex number solutions of the equation x4 81 0 . Leave answers in trigonometric form.

De Moivre’s Theorem for Finding nth Roots

If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by

or rn cis ,

where

360 k

n, k = 0, 1, 2, . . ., n – 1.

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Section 8.5 Polar Equations and Graphs

Polar Coordinate System A polar coordinate system is formed by drawing a horizontal ray called the polar axis (positive x-axis). The endpoint of the ray is called the pole (origin). Points in the polar coordinate system take the form (r, θ), where r is the directed distance from the pole and θ is measured counterclockwise from the polar axis to the segment joining the pole and the point. If θ is negative, the angle is measured in the clockwise direction. If r is negative, find the location of the angle and then go backwards “r” units. Example 1 Plot the following points in the polar coordinate system. a) (2, π/6) b) (-1, π/3) c) (4, -225°) d) (-6, 0) In a rectangular coordinate system, there is a one-to-one correspondence between the points in the plane and the ordered pairs (x, y). This is not true for a polar coordinate system! Infinitely many ordered pairs correspond to each point in a polar coordinate system. Example 2 Give two other pairs of polar coordinates for the point (3, 150°).

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Converting between Polar and Rectangular Coordinates and Equations The polar coordinates (r, θ) are related to the rectangular coordinates (x, y) as follows: x = r cos θ (r, θ) ↔ (x, y) y = r sin θ

x2 + y2 = r2 or 22 yxr

x

ytan or

x

y1tan

Example 3 Give the rectangular coordinates for the point (4, -225˚).

Example 4 Give two pairs of polar coordinates for the point , where 0˚ ≤ θ < 360˚. Example 5 For the rectangular equation 2x – 3y = 6, give its equivalent polar equation and sketch its graph. Example 6 For the rectangular equation x2 + y2 = 4, give its equivalent polar equation and sketch its graph. Example 7 For the polar equation r = 2 cos θ, find an equivalent equation in rectangular coordinates and graph. Example 8 For the polar equation r = 4 csc θ, find an equivalent equation in rectangular coordinates and graph.

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Graphs of Polar Equations Lines Oblique Line: θ = “an angle” Horizontal Line: r sin θ = a Vertical Line: r cos θ = b Circle with center at pole: r = a

11 Example 9 Sketch the graph of r = 2 + 4 sin θ by hand. Identify the type of polar graph.

12 Example 10 Sketch the graph of r = 2 cos 3θ by hand. Identify the type of polar graph.

Graphing Polar Equations on a Graphing Calculator Change MODE to “Pol” (polar) and “Degree.” Enter equation at r1. IN GENERAL adjust viewing window to θmin = 0 θmax = 360 θstep = 5

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