Put in -420 and hit cosine button and you will see that the answer is .5
The hard way to find it is to get the equivalent angle that is less than 360 degrees.
Add 360 to -420 and you get -60 degrees.
-60 degrees is in the fourth quadrant.
In the fourth quadrant, the cosine is positive.
The cosine of 60 degrees is the adjacent side divided by the hypotenuse.
The adjacent side of a 60 degree triangle in the fourth quadrant is positive.
The hypotenuse of any right triangle formed by any angle angle is always positive, regardless of what quadrant the angle is in.
The cosine of -60 degrees is 1/2 and is positive.
The cosine of -60 degrees is the same as the cosine of a 60 degree angle that resides in the fourth quadrant.
In the following diagram, you should be able to see that better.
The triangle is ABC.
AB is the side adjacent to to the 60 degree angle.
The 60 degree angle is in the fourth quadrant, and is there equal to an angle of -60 degrees because it is 60 degrees below the 0 degree line.
AC is the hypotenuse of the right triangle formed by AB, BC, and AC.
the angle between AB and BC is a right angle.
The x-axis to the right of A is the 0 degree line.
The y-axis above A is the 90 degree line.
The x-axis to the left of A is the 180 degree line.
The y-axis below A is the 270 degree line.
that's for positive angles.
For negative angle, the following holds:
The x-axis to the right of A is the -360 degree line.
The y-axis above A is the -270 degree line.
The x-axis to the left of A is the -180 degree line.
The y-axis below A is the -90 degree line.
the x-axis to the right of A is also the 360 degree line.
That's because one complete rotation through the circle is equal to 360 degrees.
