00:01
This problem, you're being asked to find the tangent value of theta plus phi.
00:04
And they give you some information about these two particular angles.
00:07
So first, the cosine of theta is negative one -third, and we know this angle is in the third quadrant.
00:13
Then they tell us that the sign of phi is equal to one -fourth and that this angle is in the second quadrant.
00:19
Well, the first thing we probably need to know is, well, how do we find this sum formula? well, remember, the sum formula for tangent says if we have the tangent of theta plus five, this will be equal to the tangent of theta plus the tangent of phi divided by 1 minus the tangent of theta times the tangent of phi.
00:43
Well, the issue is we are given the cosine and sine values of these angles, but not their tangent values, which is what we need for the formula.
00:52
However, we can use the information we have in order to find this.
00:55
So let's first start with our angle for theta.
00:58
We know it's a third quadrant angle, so let's start by drawing the angle.
01:03
We know it's in the third quadrant, so here's theta, and we know the cosine is negative one -third.
01:09
Well, remember, to find cosine is the adjacent over the hypotenuse, so the adjacent side is negative 1, and the hypotenuse is 3.
01:16
Well, we can use the pythagorean theorem in order to find the opposite side, which i'll call x.
01:22
So we'd have x squared plus negative 1 squared is equal to 3 squared.
01:27
Well, negative 1 squared is 1, so we have x squared plus 1, and then we have 3 squared, which is 9.
01:33
So we'll subtract 1 from both sides, so we find that x squared is equal to 8, and then we'll take the square root of both sides, which means that x is equal to the square root of 8, or plus or minus the square of 8.
01:48
Well, remember, we can simplify the square of 8 to 2 root 2, so we have plus or minus 2 root 2.
01:54
So in this case, think about what quadrant this is in, because xx and minus 2.
01:57
Is how in downwards, we're going to use the negative of the values.
02:00
So this side would be negative 2 root 2.
02:03
So now that we know all three sides of the triangle, we can now find the tangent value.
02:07
Remember, tangent is opposite over adjacent.
02:10
So the tangent of theta would equal to opposite negative 2 root 2 over the adjacent, which is negative 1, which means the tangent of theta is equal to 2 root 2.
02:22
Okay.
02:23
Now, remember, we still have to find the angle for 5.
02:27
So we're going to use this information right here.
02:30
So first off, we know it's in the second quadrant.
02:33
So we're going to draw down in here.
02:34
So here's our phi angle.
02:36
And we know the sign value is one -four.
02:38
Remember, sign is opposite over hypotenuse.
02:41
So the opposite is one, de -jacent, or sorry, the hypotenuse is four, and we need to find x.
02:47
So again, we'll use pythagrin theorem...