WEBVTT 00:00:00.400 --> 00:00:04.590 Simplify the cosine of 180 degrees minus πœƒ. 00:00:05.190 --> 00:00:08.620 Let’s first begin analyzing this by trying to visualize it. 00:00:09.100 --> 00:00:16.110 When dealing with the unit circle, the cosine of πœƒ represents the π‘₯-values and sine of πœƒ represents the 𝑦-values. 00:00:16.540 --> 00:00:20.850 So to the left, cosine will be negative and down sine would be negative. 00:00:21.090 --> 00:00:36.810 So in quadrant one, cosine is positive and sine is positive, in quadrant two, cosine is negative, but sine is positive, in quadrant three, cosine and sine are both negative, and in quadrant four, cosine is positive, but sine is negative. 00:00:37.330 --> 00:00:40.630 So πœƒ represents an angle. 00:00:41.640 --> 00:00:43.860 So let’s think about plugging in something. 00:00:45.020 --> 00:00:49.620 In quadrant one, we will be plugging anywhere from zero degrees to 90 degrees. 00:00:50.140 --> 00:00:57.940 So if we would take any of those values and plugged them in, we would have 180 minus 90 all the way from 180 to minus zero. 00:00:58.320 --> 00:01:04.510 So we would end up in quadrant two or it’s anywhere from 90 degrees to 180 degrees. 00:01:04.840 --> 00:01:09.980 And the cosine in that quadrant is negative. 00:01:10.630 --> 00:01:13.330 Now let’s imagine we were plugging in the pink values. 00:01:13.750 --> 00:01:26.170 If we would plug in the pink values, so 180 minus 90 all the way to 180 minus 180, we would end back into the quadrant one in the yellow. 00:01:26.840 --> 00:01:29.400 So now we’ve changed the cosine again. 00:01:29.940 --> 00:01:35.040 So if we went from a negative cosine to a positive cosine, we must put a negative in front of that. 00:01:35.620 --> 00:01:43.460 And the same thing would work down at the bottom in quadrants three and four, but let’s go ahead and go by looking at this algebraically. 00:01:44.230 --> 00:01:49.430 We can first begin by separating 180 degrees into 90 and 90. 00:01:49.890 --> 00:01:58.240 And now combining the 90 degrees minus πœƒ is actually just putting it together, which is equal to negative sine of 90 degrees minus πœƒ. 00:01:58.970 --> 00:02:07.590 Since cosine of 90 plus πœƒ equals negative sin πœƒ, so this πœƒ they’re talking about is the 90 minus πœƒ. 00:02:08.120 --> 00:02:16.950 And this is equal to negative cosine πœƒ because the sin of 90 minus πœƒ is equal to the cosine of πœƒ. 00:02:17.380 --> 00:02:22.380 So since we originally had a negative sign with sine, we have to attach it with our cosine. 00:02:22.980 --> 00:02:26.860 Therefore, after simplifying, we get negative cos πœƒ.