Question

Solve the equation -2sin^2 [theta] + cos [theta] = -2. Give exact solutions.

Answer 1

I just would like some help with figuring this out. If someone can walk me through it, great. I don't want just an answer. I need to go through the steps.

Answer 2

I'm looking for an identity now to help us solve this. Give me one sec, and I'll be back!

Answer 3

Ok, here's what I came up with. If this was a "regular" polynomial, you would solve it by factoring, right? Well the same thing applies here, but you need to have either all sines or all cosines to do that. That's where our identities come in. I just wanted to make sure I had the right one before proceeding here. Are you familiar with the Pythagorean Identities in Trig?

Answer 4

Are you still there?

Answer 5

Oh geez, this could help me. Yes, I'm here.

Answer 6

Do you know any identities that include \(sin^2(\theta)\) @WeAreSpoons ?

Answer 7

Uh, so like what kind of things would that include?

Answer 8

Go to the last chapter you learned about, what are the trig identities listed there?

Answer 9

I'm sorry I sound like an idiot. I do want to learn and I learn fast. If you can take me through this, I will be very grateful.

Answer 10

ok, take a look here: http://www.purplemath.com/modules/idents.htm

Answer 11

okay so does this have anything to do with the half-angle formula? That includes sin^2 [theta] and it says that it equals 1 - cos (2[theta]) / 2

Answer 12

Do you have a half angle?

Answer 13

like here let me get a picture of what I mean

Answer 14

where I suppose that (u) in the picture is the same as theta

Answer 15

but we don't have a half angle, that won't help us

Answer 16

ohhhhhhhhhhhh okay. is there another identity that includes sin^2 theta? I'll go look

Answer 17

yup there is, it's usually the first one you learn

Answer 18

is it a pythagorean identity?

Answer 19

Just read, you shall see it

Answer 20

It practically has strobe lights dancing off of it

Answer 21

I feel so stupid.

Answer 22

I don't even know where to start to fill in the holes in what I don't know. I'm in Pre-Calc. There's no book btw

Answer 23

Don't. Tell me, which eq on that page have a sin^2 term

Answer 24

NO BOOK??? What is wrong with that teacher...

Answer 25

I shall get you a book

Answer 26

http://home.comcast.net/~cherylwilcox1/site/

Answer 27

the basic/pythagorean and the half-angle idents are the only ones I see with a sin^2 term

Answer 28

Ok tell me each of them

Answer 29

type them out here

Answer 30

sin^2(t) + cos^2(t) = 1

That's one

Answer 31

ok

Answer 32

is that the one?

Answer 33

What's the other option first

Answer 34

uh, the half angle one, but you mentioned that there is no half angle here

Answer 35

So what are your options?

Answer 36

\[-2sin^2 [\theta] + cos [\theta] = -2\]
Your eq.

Answer 37

Do you have any options?

Answer 38

This is the whole problem. I don't know if it includes any info that I haven't said.

Solve the equation \[-2 \sin ^{2} \theta + \cos \theta = -2\]

where \[0 \le \theta < 2 pi\]

Give exact solutions.

Answer 39

does that say anything else that I haven't mentioned?

Answer 40

Ok, let's restart because I think you are overly stressing right now

Answer 41

I am D:

Answer 42

So here is your problem: \[−2sin^2θ+cosθ=−2\]

Answer 43

Now, let's think about this together

Answer 44

you need to find a way to pull out theta right?

Answer 45

I guess so. If that's what needs to happen, I am learning and I will say yes because now I have some sort of handle.

Answer 46

I guess that makes sense, because the way I've seen, I guess I just associate theta with an unknown value. Is that correct in most cases?

Answer 47

yup theta is just another variable

Answer 48

so this is like solving for x, only x is replaced with the symbol theta? okay got iy

Answer 49

it*

Answer 50

yup yup

Answer 51

I'm sorry. I'm really trying ._. so yeah, we need to isolate theta

Answer 52

so now, we want to only have cosines in our equation, that means we need to swap out that sin^2 with something equal to it

Answer 53

(csc)?

Answer 54

Any ideas on something that would be equal to \(sin^2\theta\)?

Answer 55

I'm probably grasping at straws :c

Answer 56

ok, not at all but tell me why you said that, I really don't knowhow you got that

Answer 57

sin x = 1/csc x? ehhh

Answer 58

Yes, but that's not sin^2 x

Answer 59

you want specifically, sin^2 x

Answer 60

ah, so it's the half angle identity

Answer 61

x, theta, y, alpha whatever the variable doesn't matter

Answer 62

No

Answer 63

you don't have a half angle I say again

Answer 64

You have to solve another for sin^2 x

Answer 65

that's literally all I see that has the sin^2 (x)

Answer 66

look again

Answer 67

start at the beginning and just re read everything, you'll see it

Answer 68

uh okay, does it have anything to do with like, sin^2(x) + cos^2(x) = 1?

Answer 69

yup

Answer 70

OHHHHHHHH

Answer 71

is it cos^2(x) - 1 = sin^2(x) or something?

Answer 72

continue

Answer 73

I don't know where I'm going with it haha. It's coming slowly. It's currently stoped. Hint?

Answer 74

your eq is \[−2sin^2θ+cosθ=−2
\]
You want to swap something in that contains a cosine for that sin^2

Answer 75

so that would be cos^2(x) + cos(x) - 1 = -2, right?

Answer 76

wait, what did you forget?

Answer 77

combining terms or something?

Answer 78

Remember it's \[ \color{red}{(}cos^2(x) - 1 \color{red}{)}= sin^2(x)\]

Answer 79

and your original is :\[\color{red}{−2}sin^2θ+cosθ=−2\]

Answer 80

OH, so parenthesis, so it would be (cos^2(x) - 1) + (cos(x)) = -2, right?

Answer 81

uh you highlighted the -2. What happens with the -2?

Answer 82

\[\Large\rm 1-\cos^2x=\sin^2x\]Did you guys get the identity mixed up a little bit?
Maybe I misread the last few comments... hmm

Answer 83

OH. Is that right?

Answer 84

Thank's Zep, I didn't check it

Answer 85

Thank you Zep!

Answer 86

Oh that wasn't the boo boo she was point out hehe XD I see

Answer 87

There's another one spoon! Get it!!! :D

Answer 88

And you got rid of the -2 when you put it in. It doesn't go anywhere

Answer 89

oh crap! YEAH!

Answer 90

So step by step for me now, start with the original, then sub in

Answer 91

so it's

-2(1 - cos^2(x)) + (cos(x)) = -2?

Answer 92

oh haha, well I got right to the correct subbing I guess

Answer 93

Now, simplify the left

Answer 94

I'm sorry, time is tightening, I want to work as quick as I can. So, can I actually simplify it, because one of the terms has a power

Answer 95

O.o? why is there a time limit?
and simplify=get rid of some parentheses

Answer 96

cos(x)+cos(2 x)-1 = -2

and there's a time limit because my family wats me off the computer soon

Answer 97

wants*