Question

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

Answer 1

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.