Complex Numbers III - Ambitious

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Question 1: Write the modified polar form of

z = 3 (\cos 240 ° + i \sin 240 °)

using the shorthand notation

z = 3 120 °

z = 3 120 °

z = 3 240 °

z = 3 240 °

Question 2: Write the proper polar form

z = 2 (\cos 25 ° - i \sin 25 °)

using the shorthand notation

3 25 °

2 335 °

3 - 25 °

- 3 25 °

Question 3: Multiply the following:

z_{1} = x (\cos θ + i \sin θ)

and

z_{2} = y (\cos φ + i \sin φ)

z_{1} z_{2} = x y (\cos (θ φ) + i \sin (θ φ))

z_{1} z_{2} = (x + y) (\cos (θ + φ) + i \sin (θ + φ))

z_{1} z_{2} = x y (\cos (θ + φ) + i \sin (θ + φ))

z_{1} z_{2} = x y (\cos (θ + φ) - i \sin (θ + φ))

Question 4: Find

z_{1} z_{2}

, where

z_{1} = 3 (\cos 15 ° + i \sin 15 °)

and

z_{2} = 5 (\cos 63 ° + i \sin 63 °)

z_{1} z_{2} = 8 (\cos 78 ° + i \sin 78 °)

z_{1} z_{2} = 15 (\cos 78 ° - i \sin 78 °)

z_{1} z_{2} = 8 (\cos 78 ° - i \sin 78 °)

z_{1} z_{2} = 15 (\cos 78 ° + i \sin 78 °)

Question 5: Find

z_{1} z_{2}

, where

z_{1} = 4 (\cos 31 ° + i \sin 31 °)

and

z_{2} = 7 (\cos (- 12 °) + i \sin (- 12 °))

z_{1} z_{2} = 11 (\cos 19 ° - i \sin 19 °)

z_{1} z_{2} = 28 (i \sin 43 ° - \cos 43 °)

z_{1} z_{2} = 28 (\cos 43 ° - i \sin 43 °)

z_{1} z_{2} = 28 (\cos 19 ° + i \sin 19 °)

Question 6: Find

\frac{z_{1}}{z_{2}}

, where

z_{1} = x (\cos θ + i \sin θ)

and

z_{2} = y (\cos φ + i \sin φ)

\frac{z_{1}}{z_{2}} = \frac{x}{y} (\cos (θ - φ) + i \sin (θ - φ))

\frac{z_{1}}{z_{2}} = \frac{x}{y} (\cos \frac{θ}{φ} + i \sin \frac{θ}{φ})

\frac{z_{1}}{z_{2}} = \frac{x}{y} (\cos (θ - φ) - i \sin (θ - φ))

\frac{z_{1}}{z_{2}} = x y (\cos (θ - φ) + i \sin (θ - φ))

Question 7: Find

\frac{z_{1}}{z_{2}}

, where

z_{1} = 12 (\cos 87 ° + i \sin 87 °)

and

z_{2} = 4 (\cos 29 ° + i \sin 29 °)

\frac{z_{1}}{z_{2}} = 3 (\cos 56 ° + i \sin 56 °)

\frac{z_{1}}{z_{2}} = 8 (\cos 56 ° + i \sin 56 °)

\frac{z_{1}}{z_{2}} = 3 (\cos 3 ° + i \sin 3 °)

\frac{z_{1}}{z_{2}} = 8 (\cos 56 ° - i \sin 56 °)

Question 8: Find

z_{1} z_{2} z_{3}

, where

z_{1} = 2 (\cos 15 ° + i \sin 15 °)

z_{2} = 3 (\cos 16 ° + i \sin 16 °)

and

z_{3} = 4 (\cos 17 ° + i \sin 17 °)

z_{1} z_{2} z_{3} = 24 (\cos 120 ° - i \sin 120 °)

z_{1} z_{2} z_{3} = 24 (\cos 48 ° + i \sin 48 °)

z_{1} z_{2} z_{3} = 9 (\cos 48 ° + i \sin 48 °)

z_{1} z_{2} z_{3} = 24 (\cos 120 ° + i \sin 120 °)

Question 9: Find

z^{n}

, where

z = r (\cos θ + i \sin θ)

z^{n} = r^{n} (\cos n θ + i \sin n θ)

z^{n} = r^{n} (\cos^{n} θ - i \sin^{n} θ)

z^{n} = r^{n} (\cos^{n} n θ + i \sin^{n} n θ)

z^{n} = n r (n \cos θ + i n \sin θ)

Question 10: Find

z^{3}

, where

z = 3 (\cos 25 ° + i \sin 25 °)

27 (3 \cos 25 ° + i 3 \sin 25 °)

9 (\cos 75 ° + i \sin 75 °)

27 (\cos 75 ° + i \sin 75 °)

27 (\cos^{3} 25 ° + i \sin^{3} 25 °)

Test: Complex Numbers III - Ambitious