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my notes show the solutions to four complex numbers showing how arg(z) is obtained...they also show an argand diagram showing theta...there's a couple of things i don't understand so i was hoping that someone could shed some light...thank you...

(i) z = 1 + i

(ii) z = -1 + i

(iii) z = -sqrt(3) - i

(iv) z = 1 - i

for (i), arg(z) = arctan(1) = pi/4...ane the argand diagram shows theta being projected anti-clockwise within the first quadrant...

for (ii), arg(z) = arctan(-1) = pi - pi/4 = 3pi/4...the argand diagram shows theta moving into the second quadrant anti-clockwise...

for (iii), arg(z) = arctan[1/(sqrt3)] = -5pi/6...the argand diagram shows theta in the third quadrant and is projected clockwise from the x-axis...

for (iv), arg(z) = arctan (-1) = -pi/4...the argand diagram shows theta in the fourth quadrant and it is projected clockwise from the x-axis...

*okay, here's the bits i'm having trouble with...*

for part (ii), the complex number is in the second quadrant...so to find arctan (-1)...do we take the principal angle which is pi/4 and then because it's in the second quadrant, which is pi - (the related angle), do we find arctan(-1) by calculating pi-pi/4...???...

so how was arg(z) found to be -5pi/6...???...

why is theta shown to be projected anticlockwise this time...???...

so how was arctan(-1) = -pi/4...???...

theta is shown projected anticlockwise from the x-axis...why, when the first two questions show theta projected clockwise...???...

__1.__in part (i) the complex number is in the first quadrant...arg(z) = pi/4 and i guess that's straightforward...for part (ii), the complex number is in the second quadrant...so to find arctan (-1)...do we take the principal angle which is pi/4 and then because it's in the second quadrant, which is pi - (the related angle), do we find arctan(-1) by calculating pi-pi/4...???...

__2.__for part(iii), i can see that the complex number is in the third quadrant...and i know that tan there is negative and and the angle there is found by pi + (the related angle)...i know also that arctan[1/(sqrt3)] = pi/6...i know also that arctan is restricted between -pi and pi...so how was arg(z) found to be -5pi/6...???...

why is theta shown to be projected anticlockwise this time...???...

__3.__basically the same concerns as question 2...i can see that the complex number is in the fourth quadrant...in the fourth quadrant, theta is 2pi - (the related angle)...i know that arctan(1) is pi/4...so how was arctan(-1) = -pi/4...???...

theta is shown projected anticlockwise from the x-axis...why, when the first two questions show theta projected clockwise...???...