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

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Question 1133319: Express the complex number in polar form: -5 + 5i
Answer by MathLover1(20849) About Me  (Show Source):
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Express the complex number in polar form: -5+%2B+5i

We know that any complex number, a%2Bbi, can be written in modulus-argument form,
z=r%28cos%28x%29%2Bi%2Asin%28x%29%29,
where r=sqrt%28a%5E2%2Bb%5E2%29 and x satisfies sin%28x%29=b%2Fr and cos%28x%29=a%2Fr.

if given
a%2Bbi=-5+%2B+5i =>a=-5 and b=5
find r
r=sqrt%28%28-5%29%5E2%2B5%5E2%29
r=sqrt%2825%2B25%29
r=sqrt%282%2A25%29
r=5sqrt%282%29

find angle:
sin%28x%29=b%2Fr
sin%28x%29=5%2F5sqrt%282%29=1%2Fsqrt%282%29
=>sin%5E-1%281%2Fsqrt%282%29%29=45°

cos%28x%29=a%2Fr
cos%28x%29=-5%2F5sqrt%282%29=-1%2Fsqrt%282%29.......since -cos%28x%29=sin%28x%29 and cos%2845%29 is positive value, solution is in II quadrant, angle will be
180-45=135°

=>cos%5E1%28-1%2Fsqrt%282%29%29=135°

Polar form:
z=r%28cos%28x%29%2Bi%2Asin%28x%29%29
z+=5sqrt%282%29%28cos%28135%29%2Bi%2Asin%28135%29%29
z+=7.07107%28cos%28135%29%2Bi%2Asin%28135%29%29