To verify the trigonometric identity [tex]\(\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta\)[/tex], we will follow these steps:
1. Recall the trigonometric identity:
[tex]\[
\sin 3\theta = 3\sin\theta - 4\sin^3\theta
\][/tex]
2. Choose an angle [tex]\(\theta\)[/tex] to verify the identity. For simplicity, let’s take [tex]\(\theta = \frac{\pi}{6}\)[/tex].
3. Calculate the left-hand side of the identity using [tex]\(\theta = \frac{\pi}{6}\)[/tex]:
[tex]\[
\sin 3\theta = \sin \left(3 \cdot \frac{\pi}{6}\right) = \sin \left(\frac{\pi}{2}\right)
\][/tex]
By the value of standard trigonometric functions:
[tex]\[
\sin \left(\frac{\pi}{2}\right) = 1
\][/tex]
4. Calculate the right-hand side of the identity using [tex]\(\theta = \frac{\pi}{6}\)[/tex]:
- First, calculate [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}
\][/tex]
- Then, calculate [tex]\(3 \sin \theta - 4 \sin^3 \theta\)[/tex]:
[tex]\[
3 \sin \left(\frac{\pi}{6}\right) - 4 \left(\sin \left(\frac{\pi}{6}\right)\right)^3 = 3 \cdot \frac{1}{2} - 4 \left(\frac{1}{2}\right)^3
\][/tex]
Simplify further:
[tex]\[
3 \cdot \frac{1}{2} = \frac{3}{2}
\][/tex]
[tex]\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\][/tex]
[tex]\[
4 \cdot \frac{1}{8} = \frac{1}{2}
\][/tex]
Combining these results:
[tex]\[
\frac{3}{2} - \frac{1}{2} = 1
\][/tex]
So, we have:
[tex]\[
\sin 3 \theta = 1
\][/tex]
and
[tex]\[
3 \sin \theta - 4 \sin^3 \theta = 1
\][/tex]
Both sides are equal, confirming that the trigonometric identity is indeed correct for [tex]\(\theta = \frac{\pi}{6}\)[/tex]:
[tex]\[
\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta
\][/tex]
