What is the maximum value of ${\text{sin3}}\theta\ {\text{cos2}}\theta \,{\text{ + }}\,{\text{cos3}}\theta\ {\text{sin2}}\theta $?

Hint – In order to solve this problem use the formula of sin (a + b) and apply the same in the given equation and proceed to get the highest value of the term.

Complete step-by-step solution -
The given equation is ${\text{sin3}}\theta\ {\text{cos2}}\theta \,{\text{ + }}\,{\text{cos3}}\theta\ {\text{sin2}}\theta $.
As we know \[\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b\]
Putting the values $a = 3\theta \,{\text{and }}b = 2\theta $ in above equation we get the above equation as
\[
   \Rightarrow \sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b \\
   \Rightarrow \sin (3\theta + 2\theta ) = \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \\
   \Rightarrow \sin (5\theta ) = \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \\
\]
Therefore considering \[\sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \] or \[\sin (5\theta )\] will be the same.
So the maximum value of \[\sin (5\theta )\] is 1. As the maximum value of sin function is 1.
So, the correct option is A.

Note – To solve this problem we need to think that what other can be used in place of ${\text{sin3}}\theta\ {\text{cos2}}\theta \,{\text{ + }}\,{\text{cos3}}\theta\ {\text{sin2}}\theta $ and by which formula to get the answer in the easiest way. Then we have to use the values of sin since its highest value is 1 and lowest is -1. Doing this will solve your problem.