Problem $2.$ You are asked to design a diffraction grating that will disperse the first$-$order visible spectrum through an angular range of $=27\mathrm{.}0$ when white light is shone on the grating.

$($a$)$ At what angles will the first order spectrum begin and end? $($Use the identity $sin(+)=sincos+cossin$ and consider the ratio $sin\frac{+}{sin}$ to solve for $.)$

$($b$)$ How many lines per millimeter must the grating have?

$($c$)$ Will the full second$-$order visible spectrum appear in the diffraction pattern for this grating?