Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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.
### 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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.