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Arithmetic in polar and exponential formMultiplication of complex numbersConsider two complex numbers s=p(cosα+isinα) and t=q(cosβ+isinβ). Show that st=pq(cosα+β+isin(α+β))Choose integers or surds a, b, c and d and write two complex numbers in Cartesian form, s=a+ib and t=c+idMultiplying in Cartesian form:Calculate st in Cartesian form.Convert st into polar form.Multiplying in polar form:Convert s and t each into polar form.Use the polar form of s and t to calculate st in polar form.Multiplying in exponential form:Convert s and t each into exponential formUse the exponential form of s and t to calculate st in exponential form.Convert the exponential form of st into polar form.Geometric interpretation. You have two complex numbers, s and t, on an Argand plane.Using only a ruler and protractor or their equivalence within graphing software, explain how you can use the result from question 1 to locate and determine the complex number st Confirming your explanation:Plot two complex numbers, s and t, on a Cartesian plane.Use your method from part a) to plot the complex number st.Convert s and t into polar form.Use the polar form of s and t to calculate st in polar form.Plot st from the polar form to confirm your result.Division of complex numbersConsider two complex numbers s=p(cosα+isinα) and t=q(cosβ+isinβ). Show that st=pq(cosα-β+isin(α-β))Choose integers or surds a, b, c and d and write two complex numbers in Cartesian form, s=a+ib and t=c+idDividing in Cartesian form:Calculate st in Cartesian form.Convert st into polar form.Dividing in polar form:Convert s and t each into polar form.Use the polar form of s and t to calculate st in polar form.Dividing in exponential form:Convert s and t each into exponential formUse the exponential form of s and t to calculate st in exponential form.Convert the exponential form of st into polar form.Geometric interpretation. You have two complex numbers, s and t, on an Argand plane.Using only a ruler and protractor or their equivalence within graphing software, explain how you can use the result from question 1 to locate and determine the complex number st. Confirming your explanation:Plot two complex numbers, s and t, on a Cartesian plane.Use your method from part a) to plot the complex number st.Convert s and t into polar form.Use the polar form of s and t to calculate st in polar form.Plot st from the polar form to confirm your result.Powers of complex numbersConsider the complex number z=a+ib=rcosθ+isinθ=reiθUsing Cartesian form, calculate :z2z3z-1z-2z-3Using polar form, calculate :z2z3z-1z-2z-3Using exponential form, calculate :z2z3z-1z-2z-3Choose integers or surds a and b, given the complex number z=a+ib, express the complex number in Cartesian, polar and exponential form.Using Cartesian form, calculate :z2z3z-1z-2z-3Using polar form, calculate:z2z3z-1z-2z-3Using exponential form, calculate :z2z3z-1z-2z-3Geometric interpretation. You have a complex number, s, on an Argand plane.Using only a ruler and protractor or their equivalence within graphing software, explain how you can use the result from question 1b to locate and determine the complex number sn where n is an integer. Confirming your explanation:Plot a complex number, s, on a Cartesian plane.Use your method from part a) to plot the complex number s4.Convert s into polar form.Use the polar form of s to calculate s4 in polar form.Plot s4 from the polar form to confirm your result. ................
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