It is an unfortunate bit of notation, but:
\(\displaystyle \tan^{-1}(\theta)\ne\dfrac{1}{\tan(\theta)}\)
Normally an exponent of -1 means to invert multiplicatively, but when used with a function, it means another type of inverse altogether.
The notation \(\displaystyle \theta=\tan^{-1}(x)\) or equivalently \(\displaystyle \theta=\arctan(x)\) represents an angle \(\displaystyle \theta\) such that:
\(\displaystyle \tan(\theta)=x\)
The inverse tangent function returns an angle in the 1st or 4th quadrants \(\displaystyle -\dfrac{\pi}{2}<\theta<\dfrac{\pi}{2}\), so care must be taken when using it to ensure the result is in the correct quadrant. If you are told the angle \(\displaystyle \theta\) is in the first quadrant, then:
\(\displaystyle \tan(\theta)=\dfrac{2}{3}\)
\(\displaystyle \theta=\tan^{-1}\left(\frac{2}{3} \right)\approx33.69^{\circ}\)
But, if you are told the angle is in the 3rd quadrant, then you would use:
\(\displaystyle \tan(\theta)=\dfrac{2}{3}\)
\(\displaystyle \theta=\tan^{-1}\left(\frac{2}{3} \right)+180^{\circ}\approx213.69^{\circ}\)
