Series definitions for the sin and cosine functions are:
These converge .
If we differentiate these term by term we can see that:
Many other properties can be deduced from these power series.
The graph of is as shown:
We can see that sin is not an injection (domain of
) and so there is no inverse. However the function
, or
[called the cut down sine], has the graph:
and this is an injection, and has an inverse function
with domain $latex[-1,1]$ and range
.
is the unique real number (angle) between
whose sine is x. The
is symbolised by
or
.
The graph of is a reflection of
(cut down) in the line
.
Knowing
is true, you may deduce
is true.
Example: .
Knowing is true, you may not deduce
.
Example: .
Theorem
Proof
We take the positive square root because for the
Similarly we define the ‘so called’ cut down cosine – this is cosine but with the domain , and the cut down tangent with domain
. The inverses of these functions are arccos and arctan.