I am having difficulty understanding how to be within the bound for my theta. I got $\theta = 60.9453$ (first quadrant), but it needs to be between -$\pi$ and $\pi$. If I am thinking of the unit circle, then doesn't my $\theta$; need to be either in either the 3rd or 4th quadrant, so wouldn't I need to subtract $\pi$?
Edit: I forgot that we were working in radians and not degrees. A simple mistake. But I still get confused on when I need to add or subtract by $\pi$. For example, I need to find $\theta$ for z= -3+6i. I understand tan^(-1)(6/-3)= -1.1071, but why do I need to add $\pi$ since -1.1071 is between -3.14 and 3.14?
I think that what is being overlooked is that as a math student transitions between Analytical Geometry and Real Analysis (AKA Calculus), there is a great deal of confusion about what a radian is. There is also confusion about whether the domain of trig functions such as the sine, cosine, and tangent functions are dimensionless real numbers, or angles.
All angles are associated with a unit of measure, such as a degree or a radian. Here, $360^\circ = 360~$ degrees = $2\pi$ radians.
However, the OP (i.e. original poster) is (very reasonably) interpreting the stated problem, as asking for an angle that is a dimensionless Real Number in the range $(-\pi, \pi].$
The problem stated that $\theta$ must be in the range $(-\pi, \pi].$ However, this is a range of dimensionless Real Numbers. On the other hand, the OP has probably never been trained in how to treat $\theta$ as anything other than an angle.
Resolution:
In beginning Calculus, where this problem is intended to be attacked, it is to be understood that $(-\pi, \pi]$ refers to $(-\pi ~\text{radians}, \pi ~\text{radians}]$, so that what is being asked for is a dimensioned angle.
The answer that the OP derived is not
$\theta = 60.9453$
but rather
$\theta = 60.9453~$ degrees.
The OP didn't understand that the problem was poorly written, and was asking for an angle whose unit of measure is radians, rather than degrees. With this understanding, then the OP learns that he has to apply a conversion ratio similar to converting $3$ feet into $1$ yard : $360$ degrees equals $(2\pi)$ radians.
This begs the question : why was this problem poorly worded. It is because the consideration of a radian as a unit of measure for an angle is a beginning Calculus crutch that is necessarily eradicated as the student studies Calculus more deeply.
Then, the domain of trig functions become dimensionless Real Numbers, rather than dimensioned angles. Here, the term Radian stops being used to refer to a unit of measure to be applied to angles.
Instead, the term Radian starts being used to refer to the dimensionless ratio of the arc length of a portion of the arc of a unit circle (of radius $= 1$), as compared to one complete revolution of the unit circle. The arc length of one complete revolution is $2\pi$, when the radius is $1$.
This is often referred to as $2\pi$ radians. Here, the term radian is simply the dimensionless proportion of arc length to a complete revolution. Thus, (for example) since a $45$ degree angle creates an arc whose length is $(1/8)$ of a complete revolution, the corresponding dimensionless arc length is referred to as $\pi/4$ or $\pi/4$ radians.