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Start Up Day 54 1. PLOT the complex number, z = -4 +4i 2. Determine the exact value of the absolute value of z. (The distance from (0, 0)) 3. Determine the angle Θ, in standard position, of the rotation to this complex number “z”. 4. Determine the cosine and sine of Θ. OBJECTIVE: SWBAT Plot and find absolute values of complex numbers, Write the trigonometric forms of complex numbers & Multiply and divide complex numbers written in trigonometric form. SWBAT Use DeMoivre’s Theorem to find powers & nth roots of complex numbers. EQ: How are complex numbers represented graphically? How is the absolute value of a complex number determined? How can DeMoivre’s theorem be used to express powers and/or roots of complex numbers? HOME LEARNING:p.511‐513 # 2, 4, 15, 16, 29, 30, 32, 33, 35 Lesson 6-6 The Complex Plane & DeMoivre's Theorem Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane. z x yi z x y 2 Imaginary Axis z x yi z x 2 y The modulus or absolute value of z denoted by z is the distance from the origin to the point (x, y). The angle formed from the real axis and a line from the origin to (x, y) is called the argument Real of z, with requirement that 0 Axis < 2. modified for the correct quadrant 1 y tan (always between 0 x and 2) z x yi We can take complex numbers given as and convert them to trigonometric (or polar) form. y r sin x r cos Imaginary Axis z =r 1 3 x factor r out y z r cos r sin i r cos i sin The modulus of z is the same as r. Plot the complex number: z 3 i Real Axis Find the trigonometric or polar form of this number. r 3 1 2 2 4 2 5 5 tan 1 1 With an adjustment for z 2 cos i sin 3 6 6 Quad II 5 6 Use the inverse tangent—carefully-- to determine the angle! When in Doubt, Sketch it out! Imaginary Axis 1 tan but in Quad II 3 1 z =r 1 3 x y Real Axis 5 5 z 2 cos i sin 6 6 5 6 5p arg z = 6 It is easy to convert from polar to rectangular form because you just work the trig functions and distribute the r through. 5 5 3 1 z 2 cos i sin i 3 i 2 6 6 2 2 1 2 3 2 1 2 3 5 6 If asked to plot the point and it is in polar form, you would plot the angle and radius. Notice that is the same as plotting 3 i Let's try multiplying two complex numbers in polar form together. z1 r1 cos1 i sin 1 z2 r2 cos2 i sin 2 z1 z2 r1 cos 1 i sin 1 r2 cos 2 i sin 2 Look at where andwhere we ended up and r1r2 we cosstarted i sin cos i sin 1 as to what 2 2 see if you can make a1 statement happens to Must FOIL these two complex the r 's and the 's when you multiply numbers. r1r2 cos 1 cos 2 i sin 2 cos 1 i sin 1 cos 2 i 2 sin 1 sin 2 Replace i 2 with -1 and group real terms and then imaginary terms Multiply the Moduli and Add the Arguments r1r2 cos1 cos2 sin 1 sin 2 sin 1 cos2 cos1 sin 2 i use sum formula for cos use sum formula for sin r1r2 cos1 2 i sin 1 2 Let z1 r1 cos 1 i sin 1 and z 2 r2 cos 2 i sin 2 be two complex numbers. Then z1 z2 r1r2 cos1 2 i sin 1 2 (This says to multiply two complex numbers in polar form, multiply the moduli and add the arguments) If z 2 0, then z1 r1 cos1 2 i sin1 2 z2 r2 (This says to divide two complex numbers in polar form, divide the moduli and subtract the arguments) Let z1 r1 cos 1 i sin 1 and z 2 r2 cos 2 i sin 2 be two complex numbers. Then z1 z2 r1r2 cos1 2 i sin 1 2 z1z2 r1r2 cis1 2 If z 2 0, then z1 r1 cos1 2 i sin1 2 z2 r2 z1 r1 cis 1 2 z 2 r2 If z 4 cos 40 i sin 40 and w 6 cos 120 i sin 120 , find : (a) zw (b) z w zw 4 cos 40 i sin 40 6 cos120 i sin120 4 6 cos 40 120 i sin 40 120 multiply the moduli add the arguments (the i sine term will have same argument) 24 cos160 i sin160 24 0.93969 0.34202i 22.55 8.21i If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 24 through. 4 cos 40 i sin 40 z w 6 cos120 i sin 120 4 cos 40 120 i sin 40 120 6 divide the moduli 2 cos 80 3 subtract the arguments i sin 80 In polar form we want an angle between 0° and 180° PRINCIPAL ARGUMENT In rectangular coordinates: 2 0.1736 0.9848i 0.12 0.66i 3 You can repeat this process raising complex numbers to powers. Abraham DeMoivre did this and proved the following theorem: Abraham de Moivre (1667 - 1754) DeMoivre’s Theorem If z rcos i sin is a complex number, then z r cos n i sin n n n where n 1 is a positive integer. This says to raise a complex number to a power, raise the modulus to that power and multiply the argument by that power. This theorem is used to raise complex numbers to powers. It would be a lot of work to find 3i 3i 3i 3i Instead let's convert to polar form and use DeMoivre's Theorem. r 3 2 1 4 2 2 3 i 4 you would need to FOIL and multiply all of these together and simplify powers of i --- UGH! 1 but in Quad II 5 3 6 tan 1 4 5 5 3 i 2 cos i sin 2 4 cos 4 5 i sin 4 5 6 6 6 6 4 10 10 16cos i sin 3 3 1 3 i 16 2 2 8 8 3i Solve the following over the set of complex numbers: z 1 3 We know that if we cube root both sides we could get 1 but we know that there are 3 roots. So we want the complex cube roots of 1. Using DeMoivre's Theorem with the power being a rational exponent (and therefore meaning a root), we can develop a method for finding complex roots. This leads to the following formula: 2k 2k z k r cos i sin n n n n n where k 0, 1, 2, , n 1 Let's try this on our problem. We want the cube roots of 1. We want cube root so our n = 3. Can you convert 1 to polar form? (hint: 1 = 1 + 0i) 1 0 2 2 tan 0 r 1 0 1 1 0 2 k 0 2 k z k 1 cos i sin 3 3 3 3 3 , for k 0, 1, 2 Once we build the formula, we use it first with k = 0 and get one root, then with k = 1 to get the second root and finally with k = 2 for last root. 2k z k r cos n n n We want cube root so use 3 numbers here 2k i sin n n 0 2k 0 2k zk 1cos i sin 3 3 3 3 3 0 20 z0 1cos 3 3 3 0 20 i sin 3 3 , for k 0, 1, 2 1cos 0 i sin 0 1 Here's the root we 0 21 0 21 z1 1 cos i sin 3 3 3 3 2 1 3 2 1cos i sin i 2 2 3 3 0 22 0 22 3 z 2 1 cos i sin 3 3 3 3 already knew. 3 4 4 1cos i sin 3 3 1 3 i 2 2 If you cube any of these numbers you get 1. (Try it and see!) We found the cube roots of 1 were:1, Let's plot these on the complex plane each line is 1/2 unit 1 3 1 3 i, i 2 2 2 2 about 0.9 Notice each of the complex roots has the same magnitude (1). Also the three points are evenly spaced on a circle. This will always be true of complex roots. VIDEOS: http://youtu.be/MO6 qU3SCLbM http://www.youtube. com/watch?feature= player_embedded&v =MO6qU3SCLbM http://youtu.be/tAIxdEVu TZ8 On a side note, you will sometimes see the trigonometric form of a complex number rewritten is a more condensed format: For example-2(cos 30° + i sin 30°) can also be noted as: 2cis30° *Another point of interest is that r (cosθ+ isinθ) = reiθ