Question

Find the exact value of the expression:

$\sin 195°·\cos 75°$

Short answer

The exact value is $\frac{1}{2}\left[\frac{\sqrt{3}}{2}-1\right]$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\sin 195°·\cos 75°$

We know localid="1646458669499" role="math" $\sin a·\sin b=\frac{1}{2}\left[\sin \left(a+b\right)+\sin \left(a-b\right)\right]$.

Using the formula, we get,

$\sin 195°·\cos 75°=\frac{1}{2}\left[\sin \left(195°+75°\right)+\sin \left(195°-75°\right)\right]$

Step 2. Simplify the equation.

Continuing the above equation,

$$\begin{gathered} \sin 195°·\cos 75°=\frac{1}{2}\left[\sin \left(270°\right)+\sin \left(120°\right)\right] \\ \sin 195°·\cos 75°=\frac{1}{2}\left[-1+\frac{\sqrt{3}}{2}\right] \\ \sin 195°·\cos 75°=\frac{1}{2}\left[\frac{\sqrt{3}}{2}-1\right] \end{gathered}$$