http://www.ck12.org Chapter 5. Trigonometric Functions
As a general rule, the restrictions to the domain are either the interval[−π 2 ,π 2 ]or[ 0 ,π]to keep things simple. In this
case sine is restricted to[−π 2 ,π 2 ], as shown above. To find the inverse, reflect the bold portion across the liney=x.
The blue curve below showsf(x) =sin−^1 x.
The result of this inversion is that arcsine will only ever produce angles between−π 2 andπ 2. You must use logic and
common sense to interpret these numbers in context.
Example A
What is the graph off(x) =cos−^1 x?
Solution: Graph the portion of cosine that fits the horizontal line test (the interval[ 0 ,π]) and reflect across the line
y=x.
Example B
Graph the functionf(x) =2 cos−^1 (x− 1 ).
Solution: Since the graph off(x) =cos−^1 xwas done in Example A, now you just need to shift it right one unit and
stretch it vertically by a factor of 2. It intersected thexaxis at 1 before and now it will intersect at 2. It reached a
height ofπbefore and now it will reach a height of 2π.