If
y = cos−1 x,
then
cos y = x, 0 ≤ y ≤ 𝜋.
Therefore, to find
y = cos−1
−1
,
we must find the angle y whose cosine is
−1.
There are many possible angles with this cosine, but the range of
y = cos−1 x
is restricted to
, and so y must be in this interval.
If we want to find the angle y whose cosine is -1, we need to find an angle in the interval [0, π] because the range of y = cos^(-1) x is restricted to that interval.
The cosine function has a period of 2π. When we take the inverse cosine or cos^(-1) of a value, we are finding the angle whose cosine is that value. But since the cosine function repeats every 2π, there are multiple angles that have the same cosine value.
In this case, the cosine of π is -1. So we have cos(π) = -1. This means that π is an angle whose cosine is -1.
The range of y = cos^(-1) x is [0, π], so we can conclude that y = cos^(-1) (-1) = π.