Animated Solution for Mathematics - Trigonometry: $16\sin (20^{\circ })\sin (40^{\circ })\sin (80^{\circ })$ is equal to :

Welcome, future engineers! Today, we are going to unravel a problem that, at first glance, looks like a nightmare of irrational numbers. We are asked to evaluate $16\sin (20^{\circ })\sin (40^{\circ })\sin (80^{\circ })$.

If you try to punch these into a calculator, you will get a mess of decimals. But in the world of JEE Advanced, we do not use calculators; we use insight and the beauty of mathematical structure.

Look closely at the angles: $20^{\circ }$, $40^{\circ }$, and $80^{\circ }$. In mathematics, whenever you see angles that seem to cluster around a specific value, stop and look for symmetry.

Notice that $40^{\circ }$ is $60^{\circ }-20^{\circ }$ and $80^{\circ }$ is $60^{\circ }+20^{\circ }$. We have a base angle of $\theta =20^{\circ }$, and the other two angles are perfectly positioned around a $60^{\circ }$ reference axis. This is not a coincidence; it is a setup.

Our toolkit for this problem is a powerful trigonometric identity:

This identity is a secret weapon. It allows us to compress a product of three sine terms into a single sine term of triple the angle. It is elegant, efficient, and exactly what we need.

Imagine you are standing on a circle, marking these angles. You have your $20^{\circ }$ angle, with the $40^{\circ }$ angle sitting just below the $60^{\circ }$ reference line and the $80^{\circ }$ angle sitting just above it. The symmetry is perfect.

Now, we substitute $\theta =20^{\circ }$ into our identity. The expression $16\sin (20^{\circ })\sin (40^{\circ })\sin (80^{\circ })$ transforms into:

Look at how the complexity vanishes. We are left with $16\times \frac{1}{4}\sin (60^{\circ })$. The constant $16$ divided by $4$ gives us $4$.

So, we are simply calculating $4\sin (60^{\circ })$. We know that $\sin (60^{\circ })=\frac{\sqrt{3}}{2}$.

Substituting this in, we get $4\times \frac{\sqrt{3}}{2}$, which simplifies beautifully to $2\sqrt{3}$. The problem that seemed impossible is now solved with grace.