anonymous
  • anonymous
Emergency help on an Algebra II exam! Prove: sin θ - sin θ•cos2 θ = sin3 θ. You must show all work.
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  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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nincompoop
  • nincompoop
it looks like this is an exam
anonymous
  • anonymous
It's a practice exam, but it's worth a good bit of points and while I can understand the proofs written out in my lessons, this particular problem is stumping me. I've been working on these tests for hours :(
anonymous
  • anonymous
So I know that identity of sin^2theta + cos^2theta = 1, or...something pretty close to that...

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nincompoop
  • nincompoop
show me the attempts you have made.
anonymous
  • anonymous
Okay. So the sin theta should cancel, right? cos^2 theta = sin^3 theta ****(that 3 is a cube, not sin(3))**** I don't know what to do with that!
nincompoop
  • nincompoop
how does it cancel?
anonymous
  • anonymous
Bc it's just subtracted. Like 1 - 1 = 0.
anonymous
  • anonymous
sin theta - sin theta = 0.
nincompoop
  • nincompoop
no
nincompoop
  • nincompoop
sin is multiplied to cos sin θ - (sin θ•cos2 θ) = sin3 θ ORDERS of operation
anonymous
  • anonymous
Parentheses first, then
nincompoop
  • nincompoop
btw, is that \(cos^2 \theta \) or \(cos (2 \theta) \)
anonymous
  • anonymous
The first, it's squared.
anonymous
  • anonymous
Okay, so...then do we distribute the negative into the parentheses??
anonymous
  • anonymous
I'm thinking maybe we subtract sin theta from both sides to give us (sin theta * cos^2 theta) = sin^2 theta
nincompoop
  • nincompoop
is that \(sin^3 \theta \) or \(sin (3 \theta) \)
anonymous
  • anonymous
Cubed, the first one.
nincompoop
  • nincompoop
you need to start typing these things appropriately because they mean different things
anonymous
  • anonymous
I'm sorry, I didn't realize it before I posted bc I just copy and pasted it to preserve the theta symbols :(
nincompoop
  • nincompoop
like this? PROVE \(sin ~\theta - (sin~ \theta ~cos^2 \theta) = sin^3 \theta \)
anonymous
  • anonymous
There weren't any parentheses, but otherwise, yes.
nincompoop
  • nincompoop
so tell me what should be the best identity we can use?
anonymous
  • anonymous
The only one I know of is the one I typed above, the sin^2 theta + cos^2 theta = 1.
anonymous
  • anonymous
We didn't focus a lot on identities, so I suppose I may be forgetting some, but they aren't in my notes =/
nincompoop
  • nincompoop
what did you focus on?
nincompoop
  • nincompoop
let us use trigonometric expansion and make use of some algebra techniques like factoring, yes?
nincompoop
  • nincompoop
do you think you can factor the left-hand-side (LHS) of the equation for me?
anonymous
  • anonymous
Sorry! Lost connection!
anonymous
  • anonymous
I can tryyy, hold on
nincompoop
  • nincompoop
|dw:1440734256689:dw|
anonymous
  • anonymous
sin theta(1 - cos^2 theta) = sin^3 theta
nincompoop
  • nincompoop
we can skip writing the theta for now and just add it in the end so it is easier, agreed?
anonymous
  • anonymous
Okay! sin(1 - cos^2) = sin^3
nincompoop
  • nincompoop
do you have a note for your pythagorean identities? if not it is very simple \(\large sin^2 + cos^2 = 1 \) keep in mind for future reference that \(sin = y\) and \(cos = x \)
nincompoop
  • nincompoop
|dw:1440734748047:dw|
anonymous
  • anonymous
Yep, I remember that part from the unit circle section we just did.
nincompoop
  • nincompoop
you probably already learned the Pythagorean Theorem which states that the length of the longest leg \(\large r \) is equal to the square root of the sum of the two shorter legs |dw:1440734959343:dw|
nincompoop
  • nincompoop
do you think we can manipulate our identity algebraically so it would look familiar as the problem you have?
anonymous
  • anonymous
Let me see...
anonymous
  • anonymous
y(1 - x^2) = y^3 when you say cos = x and sin = y...
nincompoop
  • nincompoop
|dw:1440735114209:dw|
anonymous
  • anonymous
I'm sorry this is taking me so long to understand ._.
nincompoop
  • nincompoop
|dw:1440735291067:dw|
nincompoop
  • nincompoop
|dw:1440735446837:dw|
nincompoop
  • nincompoop
what part did I lose you?
nincompoop
  • nincompoop
|dw:1440735591245:dw|
anonymous
  • anonymous
Where did the sin^3 go?
nincompoop
  • nincompoop
that's just the left-hand-side (LHS) since the board is too small for me to fit everything in
anonymous
  • anonymous
So it's 1^2 + sin^3 on the right hand?
nincompoop
  • nincompoop
no, the right-hand-side stays the same since it is what we're trying to accomplish
anonymous
  • anonymous
...so then where is the sin^3 if you said it's not on the left hand? D:
nincompoop
  • nincompoop
what we are basically doing is trying to manipulate and re-arrange the left-hand-side of your problem so it would EQUAL to the right-hand side of the problem
nincompoop
  • nincompoop
|dw:1440735790511:dw|
nincompoop
  • nincompoop
|dw:1440735817436:dw|
anonymous
  • anonymous
Okay, I'm mostly following
nincompoop
  • nincompoop
nincompoop
  • nincompoop
from the pythagorean identity, we derived that \(1-cos^2 = sin^2 \) because from our original identity (derived from the unit circle) \(sin^2 + cos^2 = 1 \) we can re-arrange it to \(sin^2 + cos^2 \color{red}{-cos^2} =1 \color{red}{-cos^2}\) simplifying to: \(sin^2 = 1-cos^2 \)
anonymous
  • anonymous
I don't think any of this is getting through anymore, it's really late here and I'm thinking my exam has probably timed out by now ._. I'm tracking but I don't see where this is actually leading, we keep ending up back at this identity and I'm not sure what to do with it
nincompoop
  • nincompoop
then it would suffice to say that \(sin - sin~cos^2 = sin^3 \) because \(sin (1-cos^2) = sin^3 \) because the left-hand side is equivalent to \(sin (sin^2) \) therefore \(sin^3 = sin^3 \)
anonymous
  • anonymous
I'll definitely reference this when I retake. Thank you so so so much for all of your time and help <3
anonymous
  • anonymous
You were very patient, I hope it comes back to you tenfold :x
nincompoop
  • nincompoop
you're not even telling me what part you are not following
nincompoop
  • nincompoop
up to what part you are able to grasp and what part seems a little fuzzy still
anonymous
  • anonymous
I'm sorry, it's just because I don't really know where I'm lost, I'm just really tired >.<
nincompoop
  • nincompoop
come back when your brain is fully rested. there is no point in studying when your brain is not working to its potential.
nincompoop
  • nincompoop
go to sleep
anonymous
  • anonymous
Exactly :( I hope you have a good night, thanks so much for the guidance, I'm sure it'l help a lot more in the morning
anonymous
  • anonymous
You, too!

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