Find the set of values satisfying the cos of 𝑥 is equal to one-half, where 𝑥 is greater than or equal to zero and less than or equal to two 𝜋.
By sketching a CAST diagram between zero and two 𝜋 radians, we can identify which quadrants our solutions will lie. The value of cos 𝑥 is positive. Therefore, we will have a solution in the first and fourth quadrants. For there to be a solution in the second or third quadrants, the value of cos 𝑥 would have to be negative. If the cos of 𝑥 is equal to one-half, then 𝑥 is equal to the inverse cos of one-half. We know that this is equal to 60 degrees. And in radians, this is equal to 𝜋 over three. 𝜋 over three lies in the first quadrant, as shown on the CAST diagram.
Our second solution will be a reflection of this line in the 𝑥-axis. To obtain a solution between zero and two 𝜋, we need to subtract 𝜋 over three from two 𝜋. In order to add or subtract fractions, we need a common denominator. Two 𝜋 is equivalent to six 𝜋 over three. Subtracting 𝜋 over three from this gives us five 𝜋 over three. The two angles that satisfy the equation cos 𝑥 is equal to one-half between zero and two 𝜋 are 𝜋 over three and five 𝜋 over three. This can be written in set notation as shown.