00:01
So we're given that the tangent of 2 theta is equal to negative 4 thirds.
00:07
So we know that 2 theta, in order for this to be negative, that picture -wise, we know that that 2 -theta has to either be in the second quadrant or the fourth quadrant.
00:19
And we're told that 0 is less than theta is less than 90 degrees.
00:24
So if we're dealing with this setting, we know that the double angle that if i double this and multiply everything by two, that really this angle, because of that tangent being negative, that has to be a second quadrant angle.
00:46
And the angle theta will be a first quadrant angle.
00:50
Now, we want to find what the sign of the angle is, and we want to find the cosine of the angle.
00:57
And what i notice is really this angle is half.
01:00
So i want to use my half angle identity, not my double angle identity.
01:04
I want to use my half angle identity.
01:07
So i know that the sign of half of 2 theta, half of 2 theta, kind of thinking of that as x over 2, is equal to plus or minus.
01:25
However, i know what my angle, it has to be a first quarter angle, so that's going to be plus the square root of 1 minus, the cosine of the angle, which is 2 theta over 2.
01:43
And for the cosine, again, the cosine is going to have to be positive, and i'm going to use my half angle formula, and the angle i have is 2 theta over 2, and that's equal to, and it's going to be positive, square root of 1 plus the cosine of the x or the 2 theta over 2.
02:06
So i want to draw a picture of this double angle.
02:10
And like i said, we know that that angle has to be over here.
02:15
And we'll pick a point on the terminal side of the angle and we'll drop it down.
02:19
And we'll say that this side opposite has to be positive four.
02:24
And then the adjacent must be negative three...