In Exercises 9-13 you are express the given complex number, which is presented i

Question

In Exercises 9-13 you are express the given complex number, which is presented in polar form, in algebraic form: a + bi. Use exact values. 9. z = 8cis pi/2. 10. z = 4cis 3 pi/2. 11. z = 2cis 3 pi/4. 12. z = 6cis 5 pi/6. 13. z = 2cis 7 pi/3. In Exercises 14-18 you are asked to compute a product, or a ratio of two complex numbers in two ways: (i) Algebraically. (ii) Using polar representations, that is, by presenting each one of the numbers, as well as the result, in polar form. You can use the two calculations to verify you got the right answer. 14. (4 Squareroot 3 - 4i)(2 - 2 Squareroot 3 i). 15. (4 - 4 i)(2 - 2i). 16. 4 Squareroot 3 - 4 i/2 + 2 Squareroot 3 i. 17. 8 i/2 Squareroot 3 - 2 i. 18. -6 i/3 + 3i.

Explanation / Answer

12) z= 6cis5pi/6

z = 6(cos5pi/6 + i*sin5pi/6)

= 6( cos(pi - pi/6) +i*sin(pi -pi/6) )

= 6(-cospi/6 + i*sinpi/6) = 6( -sqrt(3)/2 + i*1/2)

= -3sqrt3 + 3i

13) z= 2cis7pi/3

= 2(cos7pi/3+ i*sin7pi/3)

= 2(cos(2pi +pi/3) +i*sin(2i +pi/3) )

= 2cospi/3 + 2i*sinpi/3 = 1 + i*sqrt3

15) (4 -4i)(2 -2i)

8 -8i - 8i - 8

= -16i

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