I need help with a very basic trigonometry question.

[The Student]

I have a good grasp for trigonometry except for when I try to make sense of iscolating θ. For example, if tanθ = 2/3, then θ = (tan2/3)^(-1) = 33.69... degrees. But, knowing (tanθ)^(-1) = 1/(tanθ), and I do it that way I get 1/(tan2/3) = 85.9... degrees. I just can't see what I'm doing wrong here.

 

[MarkFL]

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:

$\displaystyle \tan(\theta)=\dfrac{2}{3}$

$\displaystyle \theta=\tan^{-1}\left(\frac{2}{3} \right)+180^{\circ}\approx213.69^{\circ}$

 

[The Student]

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:

$\displaystyle \tan(\theta)=\dfrac{2}{3}$

$\displaystyle \theta=\tan^{-1}\left(\frac{2}{3} \right)+180^{\circ}\approx213.69^{\circ}$

Got it, thanks.