Question 1133319

Express the complex number in polar form: {{{-5 + 5i}}}


We know that any complex number, {{{a+bi}}}, can be written in modulus-argument form, 
{{{z=r(cos(x)+i*sin(x))}}}, 
where {{{r=sqrt(a^2+b^2)}}} and {{{x}}} satisfies {{{sin(x)=b/r}}} and {{{cos(x)=a/r}}}.


if given

{{{a+bi=-5 + 5i}}}   =>{{{a=-5}}} and {{{b=5}}}

find {{{r}}}

{{{r=sqrt((-5)^2+5^2)}}}
{{{r=sqrt(25+25)}}}
{{{r=sqrt(2*25)}}}
{{{r=5sqrt(2)}}}


find angle:

{{{sin(x)=b/r}}}

{{{sin(x)=5/5sqrt(2)=1/sqrt(2)}}}

=>{{{sin^-1(1/sqrt(2))=45}}}°


 {{{cos(x)=a/r}}}
{{{cos(x)=-5/5sqrt(2)=-1/sqrt(2)}}}.......since {{{-cos(x)=sin(x)}}} and {{{cos(45)}}} is positive value, solution is in II quadrant, angle will be

{{{180-45=135}}}°


=>{{{cos^1(-1/sqrt(2))=135}}}°


Polar form:

{{{z=r(cos(x)+i*sin(x))}}}

{{{z =5sqrt(2)(cos(135)+i*sin(135))}}}

{{{z =7.07107(cos(135)+i*sin(135))}}}