00:03
In this problem, you're asked to solve for theta in this equation on the interval 0 to 2 pi, so we're going to be solving in radians.
00:14
What i notice about this is you've got a cosine of 2 theta there, and with cosine of 2 theta, we have several identities that we could convert that to something that has just theta in it instead of 2 theta.
00:27
And so if you look in the red to the right, cosine of 2 theta is equal to 2 cosines 4 theta minus 1, or 1 minus plus.
00:34
Sings squared theta or cosine square theta minus sine's birth theta.
00:39
And i'm thinking in terms of being able to factor this.
00:43
And so if i want to be able to factor it, then if i change it to all cosines, then that should help me factor this.
00:52
And because of that, i'm going to pick the one here at the top, not these two, because those two both have sine squared.
01:03
And then i'm going to take and plug that in for my cosine of two theta.
01:09
So i'm going to have two times two cosine squared theta minus 1 minus 7 cosine of theta.
01:24
And then i'm going to go ahead and while i'm doing this step, i'm going to have to go ahead and subtract 4 from both sides.
01:28
So i have that all on the same side of the equation.
01:31
So minus 4 equals 0, because this will cancel out.
01:40
Okay, and then i'm going to go ahead and do my math here where i distribute the two into each of those terms.
01:44
So this will be 4 cosine squared theta minus 2, minus 7 cosine of theta minus 4, and then you can take and put your negative 2 and your negative 4 together.
02:17
Okay, and then once you get to this point, you just factor this the same way you would form, factor in equations in the form of ax squared plus bx plus c.
02:30
Sometimes having the cosines in there throws people off and so if you want to rewrite this as 4x squared minus 7x minus 6 equal zero and factor it that way and then plug so essentially relettin x equal cosine of theta and then just plug it back in when you're done you can do that it's just up to you however you see it better so when you're factoring, you can use the equation for factoring, or you can just remember that in this, your 4 is a, your b is negative 7, and your c is negative 6.
03:13
And then your first step, you do ac.
03:17
So that would be 4 times a negative 6, which would be a negative 24.
03:25
Actually, i don't really necessarily need to have that negative on there, so i'm going to ignore that for now.
03:36
Okay, and then look at all of your factors of 24.
03:42
So that would be 1 in 24, 2 and 12, 3 and 8, 4 and 6.
03:49
And what you need to do, remember when you're the sign, the second of the two signs is negative, then you need to be able to subtract these two terms to get your middle term.
04:02
And so do any of those subtract to be 7? 8 is 5, no, it doesn't look like it.
04:12
And so in that case, you will have to go with your equation.
04:24
So it's not coming out nicely, so it looks like we are going to have to use the quadratic formula, where x, or in this case, or theta, or cosine of theta, would equal minus b, cluster minus the square root of b squared minus 4 a c and that's all over 2a and see if we can get an answer that way so x would equal minus a negative 7 so positive 7 plus or minus the square root of b squared so negative 7 squared would be 49 minus 4 times a is 4 times c is a negative 6, all over 2 times a, so 2 times 4.
05:34
And you just need to simplify that.
05:53
So it looks like you get 145 under there, and then over 8.
06:05
And so then go ahead and get your two answers for that.
06:11
When i calculate those, i'm getting 2 .38 or negative 0 .6309393 to 224.
06:20
And so what that gives us is our x.
06:26
So remember that our x is cosine of theta here.
06:34
And so that means cosine of theta equals 2 .38 or cosine of theta equals negative 0 .630939324.
07:02
And so one of the things i immediately notice about this is the cosine of theta fluctuates between negative 1 and 1.
07:09
It can never get to be 2 .38...