00:01
All right, so for this problem, we're basically given that we have an angle theta, and that angle is located in the second quadrant of a graph, of a basically x, y, cartesian graph, and we're given that cosine of theta is going to be equal to negative 8 over 9, and we're asked to find co -secant of theta, and we're also asked to find cotangent of theta.
00:32
So these are the two things we need to find.
00:34
So the first thing we're given is that the angle is in quadrant 2.
00:40
So if you look at over here the graph i've drawn, this is going to be quadrant 1, this is quadrant 2, this is quadrant 3, and this is going to be quadrant 4.
00:50
So we know that our angle is in quadrant 2.
00:55
So the first thing we need to consider is if our angle was in quadrant 1, both sine and cosine, would both.
01:05
Both be positive, they'd both be bigger than zero.
01:08
In the second quadrant, sine is positive, but cosine is actually going to be negative, which we actually see right here, cosine, the value of cosine is negative.
01:18
And in the third quadrant, sign is negative and cosine is also negative.
01:24
And in the fourth quadrant, sine is negative, but cosine is actually positive.
01:30
So now that we have all this rules, and we know that sign is going to be positive and cosine is going to be negative, we can go ahead and start evaluating.
01:39
So, cosine of theta, or cosine, so the data is basically, we're referring to basically a triangle.
01:49
So over here, if i draw like a sample triangle, i think it'll be easier to explain.
01:53
So we have sides, a, b, and c, and then we also have angles a, b, and c.
02:00
So let's just take angle a.
02:02
So cosine of a would be adjacent over hypotenuse, is how.
02:07
How you'd calculate it, so it'd be a over c.
02:12
And sine of a, however, would be opposite over hypotony.
02:18
So the opposite angle of a here, or the opposite side of a is b.
02:23
So it would be b, which is the opposite, or the hypotenuse, which is c.
02:27
And for cosine of a, the adjacent side to the angle of a's is the side a.
02:33
And then when we look at tangent of a we actually get opposite over adjacent so the opposite of a is b and then over adjacent adjacent of angle a is a is a side of a so now we have all these rules we can apply them to what we have here so cosine of theta that we're given is equal to negative eight over nine and we know that only cosine can be negative and sign has to be positive.
03:07
And since sign is calculated with the hypotenuse as well, we can say if we assign each of these variables, we can say a is equal to negative 8.
03:16
And i know 9 could also have been negative, and it would still be negative 8 over 9.
03:21
And the reason that can't happen is because sign has to be positive.
03:25
So if 9 was negative, then c would be negative, and that's not possible because b is always going to be positive since we're in the second quadrant...