Find the quotient z1z2 of the complex numbers z1=10(cos⁡10∘+isin⁡10∘) and z2=5(cos⁡5∘+isin⁡5∘) Leave answers in polar form.

Question

Find the quotient z1z2 of the complex numbers  z1=10(cos⁡10∘+isin⁡10∘)  and z2=5(cos⁡5∘+isin⁡5∘)  Leave answers in polar form.

lovagwb · Accepted Answer

Step 1 z1z2=10(cos⁡10∘+isin⁡10∘)5(cos⁡5∘+isin⁡5∘) ⇒z1z2=2(cos⁡10∘+isin⁡10∘)(cos⁡5∘+isin⁡5∘) ⇒z1z2=2[(cos⁡10∘+isin⁡10∘)(cos⁡5∘+isin⁡5∘)×(cos⁡5∘−isin⁡5∘)(cos⁡5∘−isin⁡5∘)] ⇒z1z2=2[cos⁡10∘cos⁡5∘−isin⁡5∘cos⁡10∘+isin⁡10∘cos⁡5∘−isin⁡10∘isin⁡5∘(cos⁡5∘+isin⁡5∘)(cos⁡5∘−isin⁡5∘)] ⇒z1z2=2[cos⁡10∘cos⁡5∘+isin⁡5∘cos⁡10∘−isin⁡10∘cos⁡5∘+isin⁡10∘isin⁡5∘(cos⁡5∘+isin⁡5∘)(cos⁡5∘−isin⁡5∘)]

Lynne Trussell · Accepted Answer

Step 1  The quotient z1z2 of the complex number, and leave the answer in Polar form.  Given: The complex numbers are z1=10(cos⁡10∘+isin⁡10∘) and z2=5(cos⁡5∘+isin⁡5∘)  Concept used:  (cos⁡θ+isin⁡θ)=eiθ  Step 2  Calculation:  The quotient z1z2 can be obtained as,  z1z2=10(cos⁡10∘+isin⁡10∘)5(cos⁡5∘+isin⁡5∘)  =2e10ie5∘i  =2e(10i−5i)  =2(cos⁡5∘+isin⁡5∘)  Thus, the quotient z1z2 of the complex numbers is z1z2=2(cos⁡5∘+isin⁡5∘)