Question

Express the complex number in trigonometric form. -3i?

Answer 1

Guess it wants polar form, lol. Well, rectangular form for a complex number is:
a + bi
the polar form (trig form) is:
\[r(\cos \theta + isin \theta)\] So now there are various conversions we have to go back and forth between the two. We need two of them, but I'll list all 4 since you'll need the others for different problems

Answer 2

Rectangular to polar:

\[r= \sqrt{a ^{2}+b ^{2}}\]
\[\tan \theta = \frac{ b }{ a }\]

Polar to rectangular:

\[x = rcos \theta \]
\[y = rsin \theta \]

So we'll use the top two conversions and go from there. With me so far?

Answer 3

yes!!

Answer 4

Alright, so what is a and what is b in your problem?

Answer 5

it only says for -3i

Answer 6

Right. So if you have the form of a + bi, but you only have bi, what must a be?

Answer 7

0!

Answer 8

Bingo :P

Answer 9

So now we need theta and r so we can put it in polar form. So we must do the 1st conversion to get r.
\[r = \sqrt{0^{2} + (-3)^{2}} = ?\] So that gives me?

Answer 10

3 haha

Answer 11

haha, right. So r is 3. Now we need theta, which weget using:
\[\tan \theta = \frac{ b }{ a }\]Problem is we justsaid a was 0, so we have:
\[\tan \theta = \frac{ -3 }{ 0 }\], which is undefined. So basically, where is tangent(theta) undefined?

Answer 12

so is it 0?

Answer 13

No, tan0 is 0, its defined there.

Answer 14

i get everything up to this point :/

Answer 15

Well:
\[\tan \theta = \frac{ \sin \theta }{ \cos \theta } \]
So if we want this to be undefined, we want values where cos(theta) = 0, that way we get 0 in the denominator.

Answer 16

cos = 0 at 270

Answer 17

and 90

Answer 18

Yep, its 0 at 90 and 270. But we know it's 270 because we have 0 -3i.
0 - 3i is the same as (0,-3), which means we are at y = -3, which definitely means 270. So 270 or 3pi/2 is your theta, giving us:
\[3(\cos270 + isin270)\] And that's your answer :3

Answer 19

ahh thank you! do you think you can help me with three more problems? they're all like this i just wanna make sure i'm right

Answer 20

Lol, alright then xD

Answer 21

Express the complex number in trigonometric form.

-3 +3rad3 i

Answer 22

so i did the first part and got 3rad3

Answer 23

For your r?

Answer 24

yes!

Answer 25

what do i do from there?

Answer 26

Sorry. And you should get 6 for the radius xD

Answer 27

ugh typical i messed that up okay an then what do i do again? :(

Answer 28

Alright, so r = 6. Now we need to do:
\[\tan \theta = \frac{ b }{ a }\] So foryou, you have:
\[\tan \theta= \frac{ 3\sqrt{3} }{ -3 }\] So now you just need to find the angle of tangent that equals that.

Answer 29

how do i figure that out

Answer 30

Well, it simplifies to -sqrt(3), right?

Answer 31

oh crap yes

Answer 32

Mhm. Well that's actually a clean value for tangent xD

Answer 33

so would it be 5pi/6

Answer 34

answers are in radians

Answer 35

Not quite. Correct quadrant, though :P

Answer 36

other option is 2pi/3 lmao

Answer 37

Haha, well if you have multiple choice then that makes it easy, eh?

Answer 38

yep!! okay two more than i promise i will let you be haha

Answer 39

Fair enough xD

Answer 40

Write the complex number in the form a + bi.

5/2(cos 150° + i sin 150°)

Answer 41

These are actually easier. What is the cosine and sine of 150?

Answer 42

-rad3/2 and 1/2

Answer 43

Okay, so now multiply both of those by (5/2)

Answer 44

-5rad3/4, 5/4 oh wow that's so easy

Answer 45

Just make sure you have an "i" on the 5/4 part. That i doesnt disappear xD

Answer 46

it's multiple choice :) okay last one is different so if you dont know how to do it its alll good

Answer 47

Alright, haha. I hope I do xD

Answer 48

i'm sure you know though

Answer 49

never though

Answer 50

Two forces with magnitudes of 25 and 30 pounds act on an object at angles of 10° and 100° respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Answer 51

never know*

Answer 52

Tricky :P But yeah, might be better if I draw this.

Answer 53

yeah i had another one like this but it was waayy easier

Answer 54

Hmm....nvm, no drawing, not sure how to draw it, haha.

Answer 55

haha if u can't get it it's alright

Answer 56

Okay, so basically, if you push on an object directly with 25 pounds of force, the object will feel the full 25 pounds of force. But if you push with 25 pounds of force at an angle, the object only receives a fraction of that force. You put more energy in than the object takes. So depending on the angle, the amount of power is reduced. In order to find out how much of a force is actually being felt, we need the polar equations we've been using. So for example, we have a 25 pound force at a 10 degree angle. This means the force that the object actually feels is 25(cos(10) + sin(10))

Answer 57

So what I am going to do is mark cosine with the variable i and sin with the variable j (because usually i is x-direction and j is y-direction with vectors). So the way to solve this is to add the two equations. The first equation I just gave:
\[25(\cos(10)i + \sin(10)j)\] The 2nd force equation is
\[30(\cos(100)i + \sin(100)j)\] Now don't do the calculator work or anything, just add those two equations together
\[25(\cos(10)i+\sin(10)j) + 30(\cos(100)i + \sin(100)j)\]

Answer 58

Well, guess there is no way to add them, the angles aren't the same. Lol, just multiply it out and such, haha.

Answer 59

And don't drop the i and j o.o

Answer 60

multiply them or add?

Answer 61

Well, distribute the 25 and the 30. Then it's all calculator work. You'll have 2 things that have the variable i and two things with the variable j that you'll combine because of like terms.

Answer 62

is it 704

Answer 63

What happened to i and j? xD

Answer 64

hahaha ahhh idk :(

Answer 65

this one isn't even multiple choice and we have to show work it sucks

Answer 66

Okay, so let me distribute everything and we'll go from there.

\[25\cos(10)i + 25\sin(10)j + 30\cos(100)i + 30\sin(100)j\]
Put those into the calculator and then combine the i terms and the j terms.

Answer 67

so 53.3?

Answer 68

53.3 what? I still don't see i's and j's xD

Answer 69

53.3 i hate this problem how about that? hahah

Answer 70

What's 25cos(10)?

Answer 71

24.620

Answer 72

Okay, so we have 24.62i
Remeber the i is there still.
Okay, 25sin(10)?

Answer 73

4.34j

Answer 74

Awesome. 30cos(100)?

Answer 75

-5.2i

Answer 76

And finally 30sin(100) :P

Answer 77

29.54j

Answer 78

Okay, now we combine like terms:
24.62i - 5.2i
4.34j + 29.54j

Answer 79

19.41i and 33.88j

Answer 80

That's your answer. In the x-direction, the force felt is only 19.41 pounds, but in the y-direction 33.88 poundsof force is felt.




So this would be the overall direction of the force.