00:01
To find the exact expressions of the trigonometric functions for the angles that have these points on their terminal side, it'll be advantageous to us to draw the points on a set of axes and to sketch that angle.
00:17
So, for example, the first one has the point negative 5 and 12.
00:21
It doesn't have to be perfect, but we know that negative 5 is going to be in the negative x direction, and then 12 will be in the positive y direction.
00:29
So let's say that that point is right there.
00:34
That is on the terminal side of the angle.
00:38
And so if the angle is in standard position, then let's say that that is the angle theta.
00:44
Now, in order to find the trigonometric ratios of theta, we would want to draw our reference right triangle.
00:51
So if you draw a right triangle straight down to the bottom of the x -axis here, that is a right angle.
00:59
And we can find the sides of this right triangle and use sine, cosine, and tangent to find what the trigonometric ratios of the angle theta are.
01:10
So the sides of this right triangle, this side is negative five because it went to the left negative five units.
01:17
And this side is positive 12 because it went up 12 units.
01:21
We also got to find the hypotenuse.
01:23
And if you use the pythagorean theorem and do this squared plus that squared, and then take the square.
01:30
Root to get that side right there, you will get that this side is 13.
01:36
So now we have our reference triangle with each side labeled.
01:41
So now we can do sine, cosine, and tangent of this reference triangle, and that'll tell us the sine, cosine, and tangent of the angle theta there.
01:52
So the sign of theta is the opposite over hypotenuse, so 12 over 13.
02:00
The cosine of theta is the adjacent over hypotenuse, so negative 5 over 13.
02:08
And the tangent of theta is going to be the opposite over adjacent, so negative 12 over 5.
02:19
They want the six trigonometric ratios, so we've also got to do the co -secant, secant, and cotangent.
02:27
So the co -sicent of theta, you just flip.
02:31
The numerator and the denominator of the sign.
02:34
So that is 13 over 12.
02:38
Secant is the reciprocal of cosine, so it's negative 13 over 5.
02:45
And then cotangent is the reciprocal of tangent, so it's negative 5 over 12.
02:52
Now we'll do that same process with part b...