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.