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Unformatted text preview: que. 846 Applications of Trigonometry Hence, the proof really boils down to showing 1
w = 1
|w | . This is left as an exercise. Next, we characterize the argument of a complex number in terms of its real and imaginary parts.
Theorem 11.15. Properties of the Argument: Let z be a complex number.
• If Re(z ) = 0 and θ ∈ arg(z ), then tan(θ) =
• If Re(z ) = 0 and Im(z ) > 0, then arg(z ) = Im(z )
Re(z ) .
2 + 2πk : k is an integer . • If Re(z ) = 0 and Im(z ) < 0, then arg(z ) = − π + 2πk : k is an integer .
• If Re(z ) = Im(z ) = 0, then z = 0 and arg(z ) = (−∞, ∞).
To prove Theorem 11.15, suppose z = a + bi for real numbers a and b. By deﬁnition, a = Re(z ) and
b = Im(z ), so the point associated with z is (a, b) = (Re(z ), Im(z )). From Section 11.4, we know
that if (r, θ) is a polar representation for (Re(z ), Im(z )), then tan(θ) = Im(z ) , provided Re(z ) = 0.
If Re(z ) = 0 and Im(z ) > 0, then z lies on the positive imaginary axis. Since we take r > 0, we
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