How do you divide $(5+2i)/(7+i)$ in trigonometric form?

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Answer 1 · 2017-04-10T17:17:33

Division of two complex numbers in trigonometric form is defined as:

$(r_1(cos(theta_1)+isin(theta_1)))/(r_2(cos(theta_2)+isin(theta_2)))= r_1/r_2(cos(theta_1-theta_2)+isin(theta_1-theta_2))$

Given: $(5+2i)/(7+i)$

We need to convert the dividend and the divisor into trigonometric form.

$r_1=sqrt(a_1^2+b_1^2)$

$r_1=sqrt(5^2+2^2)$

$r_1=sqrt(25+4)$

$r_1=sqrt(29)$

Because the signs of "a" and "b" are in the 1st quadrant, we use the equation:

$theta_1 = tan^-1(b_1/a_1)$

$theta_1=tan^-1(2/5)$

Note: We would have add $pi$ for the 2nd or 3rd quadrant and $2pi$ for the 4th.

$r_2=sqrt(a_2^2+b_2^2)$

$r_2=sqrt(7^2+1^2)$

$r_2=sqrt(49+1)$

$r_2=sqrt(50)$

Because the signs of "a" and "b" are in the 1st quadrant, we use the equation:

$theta_2 = tan^-1(b_2/a_2)$

$theta_2=tan^-1(1/7)$

$(5+2i)/(7+i)=$

$(sqrt(29)(cos(tan^-1(2/5))+isin(tan^-1(2/5))))/(sqrt(50)(cos(tan^-1(1/7))+isin(tan^-1(1/7))))=$

$sqrt(29/50)(cos(tan^-1(2/5)-tan^-1(1/7))+isin(tan^-1(2/5)-tan^-1(1/7)))$

How do you divide (5+2i)/(7+i) (5+2i)/(7+i) in trigonometric form?