Video Transcript
Is cos of 400 hundred degrees positive or negative?
To answer this question, itโs helpful to recall the unit circle. Remember, this circle has a radius of one, and we can add the following measurements of ๐ to our graph by moving in an anticlockwise direction.
We start along the ๐ฅ-axis at zero degrees. Here ๐ has a value of 90 degrees. On half a turn, itโs got a value of 180 degrees, 270 degrees, and when we get back to the start weโve done a full turn, 360 degrees.
Now the question is asking us to establish whether the value of cos of 400 degrees is positive or negative. So weโll need to continue moving in an anticlockwise direction at intervals of 90 degrees to find the relevant quadrant that 400 degrees lies in.
360 plus 90 gives us 450 degrees. The next value of ๐ is 450 degrees. This means then that 400 degrees lies somewhere in this first quadrant. So weโll need to decide whether for values of ๐ in the first quadrant cos is positive or negative.
Remember, cos ๐ is equal to adjacent over hypotenuse. So letโs give the ordered pair that corresponds to an angle of 400 degrees a name. We can call it ๐, ๐, where both ๐ and ๐ must be positive numbers since itโs in the first quadrant.
We can then construct a right-angled triangle from this ordered pair. The height of this triangle corresponds to the value of the ๐ฆ-coordinate; itโs ๐. And the width of the triangle corresponds to the value of the ๐ฅ-coordinate; itโs ๐. And of course, weโve already said that the radius of this circle has a length of one unit, so the hypotenuse of our triangle is one unit. The adjacent side in this triangle is the side immediately next to the angle ๐. Itโs got a length of ๐. And the hypotenuse is the side directly opposite the right angle. Thatโs got a length of one.
So for a value of ๐ that lies in the first quadrant, cos of ๐ is equal to ๐ over one, which is simply ๐. Since we said that ๐ must be greater than zero โ itโs a positive value โ that means that cos of ๐ must also be greater than zero since itโs equal to ๐. And thatโs for all values of ๐ that lie in the first quadrant. Since 400 degrees lies in the first quadrant, that must mean that cos of 400 degrees is positive.