12. write 18(cos169° + isin169°) in rectangular form. round numerical entries in the answer to two decimal places.

The rectangular form is z = -17.67 + i 3.43

Step-by-step explanation:

* Lets explain how change the modulus form to the rectangular form

- The rectangular form of a complex number is given by

  z = a + bi , where

  a = r cos Ф and b = r sin Ф

- The modulus form of the complex number is

  z = r(cos Ф + i sin Ф) where

  Ф =

* Lets solve the problem

∵ z = 18(cos(169)° + i sin(169)°)

∵ z = r(cos Ф + i sin Ф)

∴ r = 18 and Ф = 169°

∵ z = a + ib , where

   a = r cos Ф and b = r sin Ф

∴ a = 18 cos(169)°

∴ b = 18 sin(169)°

- Angle Ф lies in the 2nd quadrant (90° < Ф < 180°)

∵ sin(169)° is positive ⇒ (sine in the 2nd quadrant positive)

∵ cos(169)° is negative ⇒ (cosine in the 2nd quadrant is negative)

∴ a = -17.67

∴ b = 3.43

∴ z = -17.67 + i 3.43

* The rectangular form is z = -17.67 + i 3.43

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step-by-step explanation:

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: