"Solve cos(3x +20) = 0.6 for 0 < x < 360" - why are there more than one solution, and how do I find all of them?

The important part of this question is to really understand what the cosine function looks like, many students will use their calculators to find arccos(0.6) = 53.13 degrees, but not find the other solutions to the problem, eg: arcos (0.6) = 53.13, 306.37, 413.13 and so on.

Many students will therefore arrive at 53.13 = 3x+20 as the only solution and therefore 33.13 = 3x and finally x = 11.04 as the only solution, when in fact there are many.

Therefore it is always useful to draw a graph of the cosine function and demonstrate why there are multilpe solutions. We then need to know how many of the solutions will fall within the condition 0 < x <360, as a trick, we consider 0 < 3x < 1080 and then 20 < 3x + 20 < 1100. From this, we infer that any value of arccos(0.6) which falls between 20 and 1100 will be a solution to the problem.

Eg: cos(773.13) = 0.6     therefore letting 773.13 = 3x+20 we solve for x and get x = 251.04.