00:01
Alright, so we have 12 cosine squared of w minus 5 sine of w minus 10 is equal to 0.
00:15
And we want our solutions to be between 0 and 2 pi.
00:20
So this is probably where your problem is.
00:22
You have a negative solution in there, so that is not one of the answers.
00:28
It's not within our range.
00:30
So you're right, there should be four answers.
00:33
But you have negative 0 .73.
00:36
That's in the fourth quadrant, so we need to figure out what that value is in our range.
00:43
So you probably solved this by doing 12 times 1 minus sine squared of w using the pythagorean identity minus 5 sine w minus 10 equals 0.
01:03
Then we can distribute this 12 and get 12 minus 12 sine squared of w minus 5 sine w minus 10 equals 0.
01:16
Then we can combine like terms, and i'm also going to divide by a negative so that our leading coefficient is not negative.
01:25
So we'll have 12 sine squared of w minus plus 5 sine of w, and then we have negative 12 plus 10.
01:38
So we'll have negative 2 equals 0.
01:42
And from here we can factor.
01:44
It's not quite immediate what this factors to, but we should get 4 sine w minus 1 and 3 sine w plus 2 equals 0.
02:06
And how you can figure that out is look at this 12 and say, well 12 can split into 12 and 1, 6 and 2, or 3 and 4, and then negative 2 can be negative 2 and positive 1, or positive 2 and negative 1.
02:22
Then you look at those combinations and say, well which one gives me the positive 5? and it ends up being 4 and 3 and negative 1 and positive 2.
02:34
So from here we'll set both these equal to 0.
02:37
So we have 4 sine w minus 1 equals 0.
02:43
So then we get sine of w is equal 1 4th.
02:50
And then from here we can take the sine inverse of 1 4th to give us w.
02:57
So when we do that, so sine inverse of 1 4th, that's going to give us w is equal to 0 .25, which is one of the answers you had.
03:12
And then we remember, i always learned all students take calculus to remember where sine was positive.
03:23
So we have the one in the first quadrant...