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PreCalculus 2 Polar Graphs and the Complex Plane 9-3 I. Graphing Points: Complex Numbers Complex Plane - Real axis(x) and Imaginary axis(y) A. 2 + 3i B. 4 - 5i C. -3 + 5i z x yi is a complex number. The magnitude or modulus of z . Denoted by z , is defined as the distance from the origin to the point (x,y). That is z x 2 y 2 Magnitude of z is called the absolute value of z . The conjugate z x yi Therefore: z zz II. Complex Numbers: Rectangular Form vs. Polar Form Draw the triangle including x, y, r , and ! If r 0 and 0 2 , the complex number z x yi may be written in polar form as z x yi (r cos ) (r sin )i rcis is called the argument of z . Remember: r x2 y2 tan y x Plot the complex number in the complex plane and write it in polar form. Express the argument in degrees: 1 i 3i 5 7i Write each complex number in Rectangular Form 6(cos 30° + i sin 30°) 5 5 2 cos i sin 6 6 III. Product-Quotient Rule of Complex Numbers: Product Theorem Quotient Theorem (r1cis1 )(r2 cis 2 ) r1r2 cis 1 2 r1cis1 r1 cis 1 2 r2 cis 2 r2 Example: Find the product and quotient for each and write it in polar form. z 8cis20 w 4cis10 find z w find z w z 1 i w 1 3i find z w z w IV. Power Rule: De Moivre’s Theorem A. Theorem [ r( cos θ + i sin θ)] n = rn (cos nθ + i sin nθ) B. Examples: Write each expression in the standard form a+bi 1. [ 2 (cos 135° + i sin 135°)]4 2. (1 + i 3 )3 V. The “Nth” Root Theorem nth Root Theorem Put in polar form : n r k Z k n r (c is 2 k n ) and k 0, 1, 2, ..., n 1 and n the number of distinct roots 1. The complex cube roots of 8 8i 2. Find the complex 4th roots of 3 i