What is the trigonometric form of $(3+5i)$ ?

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Answer 1 · 2015-12-27T17:14:19

$3+5i = sqrt(34)(cos(tan^(-1)(5/3)) + isin(tan^(-1)(5/3)))$

$~~ sqrt(34)(cos(59.04^@) + isin(59.04^@))$

Any complex number $z = a+bi$ has a trigonometric form

$z = r(cos(theta) + isin(theta))$

where $r = |z| = sqrt(a^2 + b^2)$ and $theta = tan^(-1)(b/a)$

For the given complex number, we have $a = 3$ and $b = 5$ . Thus

$r = sqrt(3^2 + 5^2) = sqrt(9+25) = sqrt(34)$

and

$theta = tan^(-1)(5/3) ~~ 59.04^@$

So we have the trigonometric form

$3+5i = sqrt(34)(cos(tan^(-1)(5/3)) + isin(tan^(-1)(5/3)))$

$~~ sqrt(34)(cos(59.04^@) + isin(59.04^@))$

What is the trigonometric form of (3+5i) (3+5i) ?