Answer to Question #303045 in Algebra for Pyun

Question #303045

z1=3.45∠980°

z2=-5+9i

  1. z1+z2 final answer in polar form
  2. z1-z2 final answer in trigonometric form
  3. z2*z1 final answer in exponential form
  4. z1/z2 final answer in rectangular form
1
Expert's answer
2022-03-02T08:04:34-0500
"z_1=3.45\\angle980\\degree=3.45e^{i980\\degree}=3.45e^{-i100\\degree}"

"=3.45(\\cos(-100\\degree)+i\\sin(-100\\degree))"

"=3.45(\\cos(5\\pi\/9)-i\\sin(5\\pi\/9))"

"=3.45\\cos(5\\pi\/9)-3.45\\sin(5\\pi\/9)i"


"z_2=-5+9i=\\sqrt{106}e^{i\\tan^{-1}(9\/-5)}"

"=\\sqrt{106}e^{-i\\tan^{-1}(1.8)}"

"=\\sqrt{106}(\\cos(\\tan^{-1}(1.8))-i\\sin(\\tan^{-1}(1.8)))"

"=\\sqrt{106}\\cos(\\tan^{-1}(1.8)-\\sqrt{106}\\sin(\\tan^{-1}(1.8))i"

1.


"z_1+z_2=3.45\\cos(5\\pi\/9)-3.45\\sin(5\\pi\/9)i"

"+(-5+9i)=(3.45\\cos(5\\pi\/9)-5)+"


"+(-3.45\\sin(5\\pi\/9)+9)i"

"\\approx-5.60+5.60i=5.60\\sqrt{2}(\\cos(135\\degree)+i\\sin(135\\degree))"

"z_1+z_2=5.60\\sqrt{2}(\\cos(135\\degree)+i\\sin(135\\degree))"

"=5.60\\sqrt{2}\\angle135\\degree=5.60\\sqrt{2}\\ cis(135\\degree)"

2.


"z_1-z_2=3.45\\cos(5\\pi\/9)-3.45\\sin(5\\pi\/9)i"

"-(-5+9i)=(3.45\\cos(5\\pi\/9)+5)+"


"+(-3.45\\sin(5\\pi\/9)-9)i"

"\\approx4.40-12.40i=13.16(\\cos(-70.5\\degree)+i\\sin(-70.5\\degree))"

"z_1-z_2=13.16(\\cos(-70.5\\degree)+i\\sin(-70.5\\degree))"

"=13.16\\sqrt{2}\\angle289.5\\degree=13.16\\sqrt{2}\\ cis(289.5\\degree)"

3.


"z_1\\cdot z_2=3.45e^{-i100\\degree}\\cdot\\sqrt{106}e^{-i\\tan^{-1}(1.8)}"

"=3.45\\sqrt{106}e^{-i(100+\\tan^{-1}(1.8))}"


"=35.52e^{-i161\\degree}"



4.


"z_1\/z_2=3.45e^{-i100\\degree}\/(\\sqrt{106}e^{-i\\tan^{-1}(1.8)})"

"=3.45\\sqrt{106}e^{i(-100+\\tan^{-1}(1.8))}"


"=0.3351e^{-i39.0546\\degree}"

"=0.26-0.21i"


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