- #1
naav
- 18
- 0
Hi...
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...
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...?...
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...
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...?...