Express the complex number -3+ i in trig form?

Question

with angle between 0,2pi.
the answer is sqrt10cis(arctan[-1/3]+pi)
i got the same thing but pi-arctan[-1/3], are they the same? why?

how is -arctan-1/3 the same as arctan-1/3?

ok...they arnt then please explain your answer

nicee thanks

Answer 1

not the same.
when you do arctan[-1/3]
it will give you the angle in QIV, but we want the angle with the same tangent in QII. That means they are pi apart so we ADD pi to our answer

the reference angle would be arctan[1/3]
so using your subtraction method, it would be
pi - arctan(+1/3)

_______________
or you can think of it as arctan(-1/3) will be a negative value.
so if we want to go to QII, we must ADD the value.
If we subtract, like you did, then we are subtracting a negative, which equates to adding, which then lands in wrong QIII


hope that helps.

=]

..

Answer 2

Let r = - 3 + i where r is a vector in 2nd quadrant.
|r| = √((-3)² + 1²)
|r| = √10
angle = tan^(-1) (- 1 / 3)
angle = π - tan^(-1)(- 1 / 3) = 2.82 radians

PS
Let A = tan^(-1)(- 1 / 3)
A is in 2nd or 4th quadrant (tan is - ve)
For argument sake, work in degrees:-
A = 180 - 18.4° or A = 360 - 18.4°
BUT - 3 + i is in 2nd Quadrant so answer is A = 180° - 18.4°
or A = π - tan^(-1)(- 1 / 3)-----in radians
Which is your answer (so don`t bother about configuration of "other" answer)

Answer 3

5?3 - 5i r = ?((5?3)² + (-5)²) = 10 ? = atan(-5/(5?3)) = 11?/6 10(cos(11?/6) + i sin(11?/6)) answer: 10(cos(11?/6) + i sin(11?/6)) ........ FYI: is additionally written as 10 e^(11?i/6)