Question

Convert to Trigonometric Form, Round to hundredth degree. 7-24i (Please show steps to help me understand)

Answer 1

ok you are going to write this as
\[r(cos(\theta)+i\sin(\theta))\]

Answer 2

you need r and
\[\theta\]

Answer 3

yeah rcis

Answer 4

r is the easy part.

Answer 5

\[r=\sqrt{7^2+24^2}\] whatever that is

Answer 6

25

Answer 7

use a calculator if you want a decimal

Answer 8

oh ok then that was really easy

Answer 9

now to find
\[\theta\] use
\[\tan(\theta)=\frac{b}{a}=\frac{24}{-7}\]

Answer 10

-73.73

Answer 11

put your calculator in degree mode and find
\[\tan^{-1}(-\frac{24}{7})\]

Answer 12

-73.74* my bad

Answer 13

that wasn't too bad

Answer 14

i will take your word for it. if the negative angle offends you you can add 360 degrees to it to get a positive number, but your answer is correct

Answer 15

steps clear?

Answer 16

so i add 360 to it? cause the book answer is 286.26

Answer 17

there are an infinite number of ways to write it, so i guess that is what the book did, add 360

Answer 18

there is only one thing you have to be careful of. you have to know what quadrant you are in

Answer 19

so i will always use tangent?

Answer 20

let me be precise.

Answer 21

it is
\[\tan(\theta)=\frac{b}{a}\] always but that doesn't always mean that
\[\theta = \tan^{-1}(\frac{b}{a})\]

Answer 22

because arctan only gives you answers in quadrants 1 and 4, not in 2 or 3

Answer 23

in this case we were in quadrant 4 because it was 7-24i, 4 over, down 24

Answer 24

hm what would i have to do for Q2 and 3?

Answer 25

but if you are in quadrant 2 or 3 you cannot just take the arctangent. you have to adjust to make sure you are in the right quadrant

Answer 26

well suppose you wanted to do this with
\[-7+24i\] everything is still the same

Answer 27

r is still the same, and arctangent is still the same. but you are in quadrant 2 so 286.26 is obviously wrong

Answer 28

you would minus 180 right?

Answer 29

so if you got out your calculator and got -73.74 for your angle, you would have to add 180 degrees

Answer 30

yes. add or subtract. either way you got it

Answer 31

106.26

Answer 32

ah i see

Answer 33

ok i believe you. but it is clear yes? because arctan is confined to -90 to 90 so you have to adjust if you are not in quadrant 1 or 4

Answer 34

hope this helps because the steps are fairly simple

Answer 35

yeah i understand now thanks a bunch the book example is really weak and doesnt explain well glad i found this site

Answer 36

yw!