The first helpful thing to notice, is that the range of the cosine inverse function is between
![](https://latex.codecogs.com/png.latex?\bg_white 0 )
and
![](https://latex.codecogs.com/png.latex?\bg_white \pi)
. This means we can't do the old method of cancelling out the inverse cosine and the cosine because this would give us
![](https://latex.codecogs.com/png.latex?\bg_white \pi + \alpha )
which since
![](https://latex.codecogs.com/png.latex?\bg_white \alpha )
is acute would not be valid since it is outside the range.
However, we can still try to apply this by manipulating the expression. First, to give a proper definition of the trick, remember that:
This is true precisely because of the definition of an inverse function, and precisely because that domain
![](https://latex.codecogs.com/png.latex?\bg_white 0 \leq x \leq \pi )
is the domain of the original
![](https://latex.codecogs.com/png.latex?\bg_white y = \cos x )
function that we wish to 'invert'.
So proceeding from this, we want to manipulate the given expression into one in which we can use
![](https://latex.codecogs.com/png.latex?\bg_white (*) )
.
Remember that,
![](https://latex.codecogs.com/png.latex?\bg_white \cos (\pi + \alpha) = -\cos \alpha )
and
![](https://latex.codecogs.com/png.latex?\bg_white \cos (\pi - \alpha) = - \cos \alpha )
, which means
![](https://latex.codecogs.com/png.latex?\bg_white \cos(\pi - \alpha) = \cos(\pi + \alpha) )
(To see this fact more clearly, imagine drawing horizontal lines in a y = cos x graph below the x-axis, and see that when the line intersects one part of the cosine graph, it intersects the opposite side,
symmetrical to
![](https://latex.codecogs.com/png.latex?\bg_white \pi )
)
So that,