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Unformatted text preview: r θ, we have cos 2θ = cos2 θ − sin2 θ and sin 2θ = 2 sin θ cos θ. 34. Consider the equation z 6 = −64. a) Find the six roots of the equation and express them in polar form. b) Convert your answers in part (a) to rectangular form. Do not use your calculators to ﬁnd the cosine and sine of the argument. Use instead the well known fact that √ π 3 π 1 cos = and sin = . 6 2 6 2 c) Indicate the positions of the six roots in the Argand diagram. 35. For each of the following equations, ﬁnd all the solutions: a) z 7 = 32 b) z 2 = 15 + 8i c) z = z 2 d) 3z = z 2 36. Let z = −1 − i. a) Draw an Argand diagram clearly indicating the positions of z and z . b) Find the modulus z . c) Find a nonnegative real number r and an angle θ satisfying 0 < θ < 2π and z = r(cos θ + i sin θ). d) Express z 3 in polar form and then indicate its position in the Argand diagram you have drawn in part (a). 37. Consider the equation z 4 = −16. a) By ﬁrst writing z and −16 in polar form, ﬁnd all the four roo...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math, Calculus

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