00:01
This problem, we're given that cosine of theta is equal to negative three -fifths, and we are dealing with the quadrant that is from 90 degrees.
00:09
The theta has to lie between 90 and 180 degrees.
00:13
To help with each of these, we're going to need these formulas here for double angle and half angle.
00:18
First thing i want to do is i want to find my sign and tangent of theta here.
00:23
To help with this, i am going to draw a very quick diagram.
00:27
Okay, since we're going to be in the second quadrant for between 90 and 180, we're going to get, for my reference triangle, something like this.
00:34
The json will be three, and then this is five.
00:37
So using my pythagorean theorem, i could do three squared plus, we'll call that y.
00:44
Three squared plus y squared is equal to five squared.
00:48
Solving that out, i'm going to get y is equal to four.
00:50
So that means, in our scenario here, it's important to recognize that if cosine is negative, three -fiths, sine of theta, is going to equal to positive four -fiths, because sign is positive in the second quadrant, and then the tangent of theta is equal to negative four -thirds.
01:15
Okay.
01:16
So for cosine of two theta, that is cosine squared minus sine squared.
01:22
Okay.
01:22
So to help with this, i'm going to do, so for part a, i'm going to do cosine of two theta is equal to, you know, i'm going to leave that with part 8, negative three -fifths squared minus four -fifth squared.
01:41
Okay.
01:42
So the cosine of 2 -theta is equal to that's going to come out to a positive 9 over 25.
01:49
This will be a 16 over 25, and then we're going to subtract them.
01:53
So the cosine of 2 -theta is equal to a negative 7 over 25.
01:58
Okay.
01:59
Part b.
02:01
Sign of 2 theta, okay, we're going to use this one, two times sine theta, times cosine theta.
02:11
So it's going to be two times the sine of theta, which is 4 fifths, times the cosine of theta, which is negative three -fiths.
02:19
So we just multiply these three fractions together, and we get negative 24 -fifths.
02:25
Part c, we got tangent of 2 -theta, okay, and that's going to use this formula.
02:33
So i'm going to have two times the tangent, which is negative four thirds, divided by one minus tangent squared.
02:44
So i'm going to take negative four thirds to the second power.
02:48
Okay.
02:48
Let's scroll down a little bit here.
02:50
This will be two times negative four thirds.
02:52
It's going to give me negative eight thirds.
02:54
Underneath, i'm going to have one minus 16 over nine.
03:00
So that's going to become negative eight thirds all over negative seven ninths...