Question

Select all angle measures for which cos 0=1/2

Answer 1

The cosine ratio is the side adjacent over the hypotenuse in a right triangle.  If we set this reference angle up, the side adjacent to the reference angle has to be where x is positive because our 1/2 is a positive value.  The hypotenuse, no matter where it lies, is ALWAYS positive.  x is positive in the first and fourth quadrants.  Therefore, the value along the x axis is 1 and the hypotenuse is 2 (cosine is adjacent over hypotenuse...giving us our 1/2).  If the side adjacent is 1 and the hypotenuse is 2, the side across from the reference angle is the square root of 3, found using Pythagorean's theorem (or your knowledge of the Pythagorean triple for a 30-60-90 triangle).  The angle that lies across from the square root of 3 is the 60 degree angle in the first quadrant or a 300 degree angle in the fourth.  But we don't have the choice of a 60 or a 300!  We measure angles usually from the positive x axis counterclockwise.  This gives us the positive angle measure.  But if we measure the angle from the positive x axis clockwise, we measure the negative angle.  A positive 300 degree angle is the same thing as a -60 degree angle.  If the positive angle in the fourrth quadrant is a 300 degree angle and we go around the whole coordinate plane one more time (which is 360 degrees) we have 300+360 which is 660.  So your choices that are correct are -60° and 660°.  Now that we did them the correct way, by placing the angles in the coordinate plane, go ahead and check them on your calculator in degree mode to see that we are correct.

Answer 2

The general solution to cos(theta) = 1/2 is 2k π ± π /3, translated to degrees, the solution is
360k ± 60 degrees, where k is an integer.

For k=0, we have solutions 0 ± 60, or  {-60, +60}
For k=1, we have solutions 360 ± 60, or  {300, 420}
For k=2, we have solutions 720 ± 60, or  {660, 780}

This is sufficient to cover the range of the answer choices, giving the angle measures for which cos(theta)=1/2 as {-60,660}