$$The point P$(-2,0)$ lies on the curve y $=$ sin $(5$x$)$

a$)$ If Q is the point $($x$,$ sin$(5$x$)),$ find the slope of the secant line PQ $($correct to six decimal places$)$ for the following values of x:

I$)0$

ii$)-2\mathrm{.}1$

iii$)-2\mathrm{.}01$

iv$)-2\mathrm{.}001$

v$)-1\mathrm{.}5$

vi$)-1\mathrm{.}9$

vii$)-1\mathrm{.}99$

viii$)-1\mathrm{.}999$

b$)$ Using the results from part a$),$ guess the value of the slope of the tangent line to the curve at P$(2,0)$

c$)$ Using the slope from part b$),$ find an equation of the tangent line to the curve at P$(2,0)$