Unformatted text preview: on in our
text.1 Instead, we use the notation f −1 (x) = arccos(x), read ‘arc-cosine of x.’ To understand the
‘arc’ in ‘arccosine’, recall that an inverse function, by deﬁnition, reverses the process of the original
function. The function f (t) = cos(t) takes a real number input t, associates it with the angle
θ = t radians, and returns the value cos(θ). Digging deeper,2 we have that cos(θ) = cos(t) is the
x-coordinate of the terminal point on the Unit Circle of an oriented arc of length |t| whose initial
point is (1, 0). Hence, we may view the inputs to f (t) = cos(t) as oriented arcs and the outputs as
x-coordinates on the Unit Circle. The function f −1 , then, would take x-coordinates on the Unit
Circle and return oriented arcs, hence the ‘arc’ in arccosine. Below are the graphs of f (x) = cos(x)
and f −1 (x) = arccos(x), where we obtain the latter from the former by reﬂecting it across the line
y = x, in accordance with Theorem 5.3.
2 π π
2 x −1
reﬂect across y = x f (x) = cos(x), 0 ≤ x ≤ π −− − − − −→
switch x and y...
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