Trigonometric functions and the unit circle

[Niles]

Homework Statement

Hi all.

Today I had to solve: $\cos \theta = -1/2$. What I did was to look in a table to find that $\theta = 2\pi/3 \quad \text{and}\quad \theta = 4\pi/3$.

My question is what is the general strategy when I wish to write this as a a function of an integer n? Is there even a general strategy for this?

Thanks in advance.

 

[CompuChip]

... when I wish to write this as a a function of an integer n? Is there even a general strategy for this?

What do you mean?
Can you solve
$$\cos\theta = n$$
without looking it up (no, unless you have a calculator), or can you write the solutions
$$\theta = 2\pi/3, 4 \pi / 3$$
as
$$\theta = \frac{2\pi}{3} + 2 \pi n, \frac{4 \pi}{3} + 2 \pi n$$
or can you combine those into one formula (no, not particularly nicely).

 

[Niles]

What I meant was your #2 suggestion: If there is any way to write the two solutions nicely for all n.

Hmm, ok then. I will just stick to writing the solutions as you did. Thanks.

 

[CompuChip]

OK, so maybe you can write something like
$$\frac{2 \epsilon \pi}{3} + 2 \pi n$$
with $n \in \mathbb{Z}, \epsilon = 1, 2$
or
$$\frac{k \pi}{3}$$
with $k \in \mathbb{Z}, k \equiv 2 \text{ or } 4 \, \operatorname{mod} 7$.

But I think the expression from my earlier post is more common (in any case, you only need to think about it half as long to understand what it's saying and where it comes from ).

 

[Niles]

Ok, thanks!