Simplify the cosine of 180 degrees minus 𝜃.
Let’s first begin analyzing this by trying to visualize it. When dealing with the unit circle, the cosine of 𝜃 represents the 𝑥-values and sine of 𝜃 represents the 𝑦-values. So to the left, cosine will be negative and down sine would be negative. 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.
So 𝜃 represents an angle. So let’s think about plugging in something. In quadrant one, we will be plugging anywhere from zero degrees to 90 degrees. 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. So we would end up in quadrant two or it’s anywhere from 90 degrees to 180 degrees. And the cosine in that quadrant is negative.
Now let’s imagine we were plugging in the pink values. 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. So now we’ve changed the cosine again. So if we went from a negative cosine to a positive cosine, we must put a negative in front of that. 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.
We can first begin by separating 180 degrees into 90 and 90. And now combining the 90 degrees minus 𝜃 is actually just putting it together, which is equal to negative sine of 90 degrees minus 𝜃. Since cosine of 90 plus 𝜃 equals negative sin 𝜃, so this 𝜃 they’re talking about is the 90 minus 𝜃. And this is equal to negative cosine 𝜃 because the sin of 90 minus 𝜃 is equal to the cosine of 𝜃. So since we originally had a negative sign with sine, we have to attach it with our cosine. Therefore, after simplifying, we get negative cos 𝜃.