Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematical Investigations IV Name: Mathematical Investigations IV Complex Concepts-Making “i” contact Products, Quotients, DeMoivre’s Theorem What happens when we multiply two complex numbers? We have observed a relationship between the radii and angles of the two complex numbers and the resulting product. Let’s prove this relationship in the general case: (r1 cis ) (r2 cis ) r1 r2 cis( ) (r1 cos + i r1 sin ) (r2 cos + i r2 sin ) Write (r1 cis )·(r2 cis ) in polar form. Multiply out the terms in the parentheses. _____________________________________ Group the real terms and the imaginary terms together. _____________________________________ Simplify, using trig identities. _____________________________________ Rewrite in cis form _____________________________________ Next, let’s divide one complex number by another and prove the analogous relationship between the radii and angles of the complex numbers and their quotient: r1 cis r1 cis( ) r2 cis r2 Write r1 cis in polar form r2 cis r1 cos + i r1 sin r2 cos + i r2 sin Rationalize the denominator ________________________________________ Multiply everything out ________________________________________ Group the real and imaginary terms ________________________________________ Use trig identities to simplify ________________________________________ Rewrite in cis form ________________________________________ Complex 2.1 Rev F06 Mathematical Investigations IV Name: 1. Fill in the blank. 6 cis = 4 5 cis 40 cis 30 = cis _____ 3cis cis 2 cis 3 = cis_____ cis 12 cis(70) cis235 = cis ______ cis (120) (5 cis 20)( ___ cis ____) = 10 cis 70 (4 cis 50)(___ cis ____) = -4 cis 50 2. For z = r cis , find the following in terms of r and z2 = _______________ z3 = _______________ z15 = _______________ zn = _______________ Generalizing from our results from zn, we state DeMoivre’s Theorem: If z = r cis , then zn = rn cis (n) 3. Use DeMoivre’s Theorem to simplify the following: (2 cis 12°)6 =________________ (4 cis 23°)3 = _______________ (cis 35 )6 (cis 20 )4 cis 25 = __________ (4 cis 16°)3 (2 cis (–10°))3 = ____________ Complex 2.2 Rev F06 Mathematical Investigations IV Name: (((cis 5°)2 cis 10°)2 cis 20°)2 = ___________ 4. Let z = 2 cis 45°. Find z2, z3, z4, and z5 in polar form. Plot z and each of these powers on an Argand diagram. 5. Let z = cis15°. Plot z, z2, z3, ... , z7 on an Argand diagram. Also locate z10, z30, and z100. z and all of the powers listed above are all solutions to the equation zn = 1 for some values of n. Find this value of n. Complex 2.3 Rev F06