Answer
(a) Slope=$1$; Number of Cycles=Slope=$1$ ;Slope$=a=1.$
(b) Slope=$2$; Number of Cycles=Slope=$2$; Slope$=a=2.$
Work Step by Step
(a) $\dfrac{d}{dx}\sin{x}=\cos{x}\rightarrow cos(0)=1\rightarrow$ Slope of the tangent line is 1.
Complete Cycle consists of two humps so graph (a) has one complete cycle.
$\sin{x}=\sin{1x}\rightarrow a=1$.
Slope=1; Number of Cycles=Slope=1 ;Slope=a=1.
(b) Using the Chain Rule: $\dfrac{d}{dx}\sin{2x}=\dfrac{d}{dx}2x\times \cos{2x}=2\cos{2x}$.
$2\cos{0}=2\rightarrow$ The slope of the tangent is $2$.
Complete Cycle consists of two humps so graph (b) has two complete cycles.
$\sin{2x}\rightarrow a=2$.
(b) Slope=2; Number of Cycles=Slope=2; Slope=a=2.
