Certainly! Let's verify the trigonometric identity [tex]\(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\)[/tex] step-by-step.
### Step 1: Express [tex]\(\sin(3\theta)\)[/tex] using a known trigonometric identity
The triple angle formula for the sine function is a well-known trigonometric identity:
[tex]\[
\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)
\][/tex]
### Step 2: Verify the identity
We'll use this triple angle formula directly:
[tex]\[
\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)
\][/tex]
The expression on the right-hand side is exactly what we have:
[tex]\[
3\sin(\theta) - 4\sin^3(\theta)
\][/tex]
So, according to the given trigonometric identity, this should be identical to [tex]\(\sin(3\theta)\)[/tex].
### Step 3: Simplify and equate
By directly applying the trigonometric identity:
[tex]\[
\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)
\][/tex]
Given that both sides are representations of the same trigonometric function, we conclude that the identity holds true.
Thus, the expression [tex]\(\sin(3\theta)\)[/tex] is indeed [tex]\(3\sin(\theta) - 4\sin^3(\theta)\)[/tex], and this verifies the trigonometric identity.
### Final Verification
As per our verification, it is clear that:
[tex]\[
\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)
\][/tex]
The identity holds true, and you can confidently use this result in your trigonometric calculations.
