Certainly! Let's solve the equation step-by-step to verify the two sides of the equation:
[tex]\[
-\frac{\pi}{3} \sin \frac{\pi}{6} = \frac{1}{2} \left( \sin \frac{\pi}{2} - \sin \frac{\pi}{6} \right)
\][/tex]
### Step 1: Calculate \(\sin \frac{\pi}{6}\)
The sine of \(\frac{\pi}{6}\) is:
[tex]\[
\sin \frac{\pi}{6} = \frac{1}{2}
\][/tex]
### Step 2: Calculate the left-hand side of the equation
Now, substitute \(\sin \frac{\pi}{6}\) into the left-hand side:
[tex]\[
-\frac{\pi}{3} \sin \frac{\pi}{6} = -\frac{\pi}{3} \cdot \frac{1}{2} = -\frac{\pi}{6}
\][/tex]
### Step 3: Calculate \(\sin \frac{\pi}{2}\)
The sine of \(\frac{\pi}{2}\) is:
[tex]\[
\sin \frac{\pi}{2} = 1
\][/tex]
### Step 4: Calculate the right-hand side of the equation
Substitute \(\sin \frac{\pi}{2}\) and \(\sin \frac{\pi}{6}\) into the right-hand side:
[tex]\[
\frac{1}{2} \left( \sin \frac{\pi}{2} - \sin \frac{\pi}{6} \right) = \frac{1}{2} \left( 1 - \frac{1}{2} \right) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}
\][/tex]
### Step 5: Compare the two sides
We have calculated the following:
- Left-hand side: \(-\frac{\pi}{6}\)
- Right-hand side: \(\frac{1}{4}\)
Now compare the two sides:
[tex]\[
-\frac{\pi}{6} \quad \text{and} \quad \frac{1}{4}
\][/tex]
Since \(-\frac{\pi}{6} \approx -0.5235987755982987\) and \(\frac{1}{4} = 0.25\), it is clear that:
[tex]\[
-\frac{\pi}{6} \ne \frac{1}{4}
\][/tex]
So, the two sides of the equation are not equal. Therefore, the given expression is false.
