There are two more solutions. Just not 223 or 583. What assumption did you make that got you those two numbers?
You really need be familiar with trig functions of special angles. The ratio can be found from the 30-60-90 special triangle. In our case,
.
Also know the cartesian plane. Cosine is positive in Quadrants I and IV, so you can find the angle related to 30 degrees in Quadrant IV by subtracting it from 360 degrees. So
is also true.
It looks like you're already familiar with coterminal angles, so between that and what I have above, you should be able to find the four solutions.
What's missing is an understanding of the symmetry of the cosine function. You can flip it about a vertical axis of , and its graph will look the same. is the smallest number such that for all So trying to find solutions based on any smaller periodicity of the cosine function isn't going to work, I'm afraid.
So, you've got one answer you know is right: 43 degrees. Flip that about the axis. What do you get?
Cosine is positive in the first and fourth quadrants.
Have you learned about reference angles? Take a look at this page from Spark Notes:
SparkNotes: Trigonometry: Trigonometric Functions: Reference Angles
I know that , and that cosine is also positive in the 4th quadrant. To find the angle whose reference angle is 30°, subtract 30° from 360° (this is true for any angle in the 4th quadrant) and you get 330°. So is also true.