Notice how the graph and mapping diagram visualize the following
facts that for the sine, cosine and tangent functions.
Theorem: TRIG.SHAPE The shape of the sine, cosine, and tangent
functions.
Increasing and Decreasing:
- The sine function is increasing for all intervals $
[(4k-1)\pi/2,(4k+1)\pi/2]$ and decreasing for all intervals $
[(4k+1) \pi/2, (4k+3) \pi/2]$ when $ k \in Z$.
- The cosine function is increasing for all intervals $
[(2k+1)\pi,2k\pi]$ and decreasing for all intervals $ [2k \pi, (2k+1) \pi]$ when $ k \in Z$.
- The tangent function is increasing of all intervals $ ((2k-1) \pi/2, (2k+1)\pi/2) $when $ k \in Z$.
Extremes:
- The sine function has maximum value of $1$ when $x = (4k+1)\pi/2$ and
minimum value of $-1$ when $x = (4k-1)\pi/2$ when $ k \in Z$.
- The cosine function has maximum value of $1$ when $x = 2k\pi$ and
minimum value of $-1$ when $x = (2k+1)\pi$ when $ k \in Z$.
The justification for these facts can be understood by considering the
mapping diagrams for the unit circle definitions of these functions as well as the more common graphical view.
On the graphs of these functions, this is
visualized by the "vertices" of the wave curves. On the mapping
diagram this is visualized by the fact that the arrows all land
above (and below) the values $-1 $ (and $1$) on the target axis.