1. Applications of Trigonometric Functions
Chapter 7
Applications of Trigonometric Functions
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2. Polar Form of Complex Numbers; DeMoivre’s Theorem
SECTION 7.8
Represent complex numbers geometrically.
Find the absolute value of a complex number.
Write a complex number in polar form.
Find products and quotients of complex numbers in polar form.
Use DeMoivre’s Theorem to find powers of a complex number.
Use DeMoivre’s Theorem to find the nth roots of a complex number. 1 2 3 4 5 6

3. GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS
The geometric representation of the complex number a + bi is the point P(a, b) in a rectangular coordinate system.
When a rectangular coordinate system is used to represent complex numbers, the plane is called the complex plane.
The x-axis is also called the real axis, because the real part of a complex number is plotted along the x-axis. Similarly, the y-axis is also called the imaginary axis.
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4. GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS
A complex number a + bi may be viewed as a position vector with initial point (0, 0) and terminal point (a, b).
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EXAMPLE 1
Plotting Complex Numbers
Plot each number in the complex plane.
1 + 3i, –2 + 2i, –3, –2i, – i
Solution
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6. ABSOLUTE VALUE OF A COMPLEX NUMBER
The absolute value (or magnitude or modulus) of a complex number z = a + bi is
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7. POLAR FORM OF A COMPLEX NUMBER
The complex number z = a + bi can be written in polar form
where a = r cos θ, b = r sin θ, and
When a nonzero complex number is written in polar form, the positive number r is the modulus or absolute value of z; the angle θ is called the argument of z (written θ = arg z).
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Writing a Complex Number in Rectangular Form
EXAMPLE 4
Write the complex number
in rectangular form.
Solution
The rectangular form of z is
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9. PRODUCT AND QUOTIENT RULES FOR TWO COMPLEX NUMBERS IN POLAR FORM
Let z1 = r1(cos 1 + i sin 1) and z2 = r2(cos θ2 + isin θ2) be two complex numbers in polar form. Then
and
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Finding the Product and Quotient of Two Complex Numbers
EXAMPLE 5
Leave the answers in polar form.
Solution
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Finding the Product and Quotient of Two Complex Numbers
EXAMPLE 5
Solution continued
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DEMOIVRE’S THEOREM
Let z = r(cos + i sin) be a complex number in polar form. Then for any integer n,
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EXAMPLE 7
Finding the Power of a Complex Number
Let z = 1 + i. Use DeMoivre’s Theorem to find each power of z. Write answers in rectangular form.
Solution
Convert z to polar form. Find r and . ,
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EXAMPLE 7
Finding the Power of a Complex Number
Solution continued
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EXAMPLE 7
Finding the Power of a Complex Number
Solution continued
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16. DEMOIVRE’S nth ROOTS THEOREM
Let z and w be two complex numbers and let n be a positive integer. The complex number z is called an nth root of w if zn = w.
The nth roots of a complex number w = r(cos  + i sin ), where r > 0 and  is in degrees, are given by
for k = 0, 1, 2, …, n – 1.
If  is in radians, replace 360º with 2π in zk.
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EXAMPLE 8
Finding the Roots of a Complex Number
Find the three cube roots of 1 + i in polar form, with the argument in degrees.
Solution
In the previous example, we showed that
and .
Use DeMoivre’s Theorem with n = 3.
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EXAMPLE 8
Finding the Roots of a Complex Number
Solution continued
Substitute k = 0, 1, and 2 in the expression for zk and simplify to find the three cube roots.
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EXAMPLE 8
Finding the Roots of a Complex Number
Solution continued
The three cube roots of 1 + i are as follows:
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