Periodicity of Inverse Trigonometric Functions

[Liger20]

Homework Statement

My problem from before has been more or less resolved, but now I have a new, bigger problem. I need to figure out how to find recuring values for trig functions. I'm having a hard time figuring out how to

1. Get the equations associated with a given value for the trig functions
2. Actually finding all of those values.

This is a problem for me because I can tell that this is a critical concept.

Homework Equations

My book gives me these equations for radians: x=30+360n or x=150+360n
For radians: pi/6+2piN or 5pi/6+2piN. The book goes into little detail as to how to work these equations, and I would very much appreciate it if someone could tell me how to work them.

The Attempt at a Solution

One problem reads: List the solutions in degrees: cos^-1(1/2)= 60 (degrees), 300, 420, 660, 780, 1020, and I have absolutely no idea how they came up with that answer. Like I said before, I would really appreciate it if someone could help me with this, and please tell me if I should clarify any part of what I'm asking.

 

[Mark44]

Think about it the other way around: What are all the angle measures (in degrees) for which the cosine of that angle is 1/2?

There are going to be two angles in each 360 degree full circle, so (in degrees),
.5 = cos(60) = cos(300) = cos(360 + 60) = cos(360 + 300) = cos(720 + 60) = cos(720 + 300) = ...

All of the cosine function arguments are of the form 60 + n*360 or 300 + n*360, where n is an integer. (I have shown them for nonnegative integers n, but the pattern applies also to negative integers.

 

[Liger20]

Whoa, that just clicked beautifully. THANK YOU! I'm pretty sure I understand what I'm doing now.

 

[Mark44]

Whoa, that just clicked beautifully. THANK YOU! I'm pretty sure I understand what I'm doing now.

You're welcome!