Stitz-Zeager_College_Algebra_e-book

# Since x t the bounds on t match precisely the bounds

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ) . π 2 + 2πk : k is an integer . • If Re(z ) = 0 and Im(z ) < 0, then arg(z ) = − π + 2πk : k is an integer . 2 • 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 z) that if (r, θ) is a polar representation for (Re(z ), Im(z )), then tan(θ) = Im(z ) , provided Re(z ) = 0. Re( If Re(z ) = 0 and Im(z ) > 0, then z lies on the positive imaginary axis. Since we take r > 0, we have...
View Full Document

## This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

Ask a homework question - tutors are online