00:01
In this problem, we want to use trigonomic entities for the double angle to find the sine of 2 theta, the cos of 2 theta, and the tan of 2 theta given the cosecant of theta and the cotangent of theta.
00:19
So we're not given theta explicitly, and we don't really need it.
00:22
So let's recall what are our double angle entities.
00:26
So the sine of 2 theta can write as 2 times the tan of theta over 1 plus the tangent squared of theta.
00:40
The cos of 2 theta writes as 1 minus the tangent squared of theta divided by 1 plus the tangent squared theta, and the tangent of 2 theta is 2 times the tan of theta divided by 1 minus the tan squared.
01:24
So there are multiple formats of a double angle formula, but there are some formulas for the sine of 2 theta, wood sines and cosines, and similar for the code of 2 theta.
01:33
But because the tangent of 2 theta is most simply written as a function of tangent, i just wrote everything as a tangent, so we only find the tangent from our given angles, and then we can easily find the rest.
01:46
So in our first scenario, we have that the cosecant of theta is equal to minus 5 over 2.
01:59
Now what does this mean? what is the cosecant? so the cosecant is 1 over the sine, so this translates as 1 over the sine theta of minus 5 over 2.
02:12
And so here we're given a constraint on the angle just because of signs.
02:20
Plus or minus signs can be a little bit tricky because because you can multiply the functions by periodic, there's often two solutions, a negative or a positive for theta ranging from 0 to pi.
02:30
So we have constraints to help us later on, but we're not there yet.
02:34
First, from here, we can solve for the sine of theta.
02:38
The sine of theta now becomes minus 2 over 5.
02:45
Now to find a tangent, we're going to need the cos of theta.
02:50
The cos of theta, we can always find it from the sin theta by using the identity 1 minus the sin squared theta.
02:57
And now we see that the cos theta we'll write as the square root of 21 over 5, given that theta, sin theta is minus 2 over 5.
03:08
And again, this is where we need to have a constraint on theta because there's two possible solutions here.
03:18
So which solution are we going to need? well, for theta ranging between 3 pi over 2 and 2 pi, what is a sign of cos theta? so cos theta looks like this.
03:33
In this regime, 3 pi over 2 to 2 pi, the cos theta is positive.
03:39
So here we're only interested in the positive solution...