To determine whether the trigonometric identity [tex]\(\sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta)\)[/tex] is true, let’s examine both sides of the equation separately and then compare them.
### Step-by-Step Solution:
#### Step 1: Define the Identity
Consider the trigonometric identity [tex]\(\sin(3\theta)\)[/tex]:
[tex]\[
\sin(3\theta)
\][/tex]
#### Step 2: Express the Left-Hand Side (LHS)
The left-hand side (LHS) of the equation is:
[tex]\[
\sin(3\theta)
\][/tex]
#### Step 3: Express the Right-Hand Side (RHS)
The right-hand side (RHS) of the equation is given by:
[tex]\[
3 \sin(\theta) - 4 \sin^3(\theta)
\][/tex]
#### Step 4: Compare the Sides
To prove the identity, we need to show that:
[tex]\[
\sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta)
\][/tex]
#### Step 5: Verification
One way to verify whether these expressions are indeed equivalent is to use trigonometric identities. Fortunately, [tex]\(\sin(3 \theta)\)[/tex] has a known trigonometric identity which states:
[tex]\[
\sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta)
\][/tex]
### Conclusion
By the known trigonometric identity, the expression for [tex]\(\sin(3\theta)\)[/tex] indeed matches the right-hand side expression [tex]\(3 \sin(\theta) - 4 \sin^3(\theta)\)[/tex]. Therefore, the identity is true, and we have:
[tex]\[
\sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta)
\][/tex]
Hence, the trigonometric identity [tex]\(\sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta)\)[/tex] is verified as correct.
