Evaluate 3sinx - 4sin 3x and figure out the correct answer from below options?
sin 3x
To evaluate the expression \(3\sin x - 4\sin^3 x\), we need to recognize a fundamental trigonometric identity. This specific form of the expression is directly related to the triple angle formula for sine.
Triple Angle Formula for Sine
The triple angle formula for sine is a key identity in trigonometry that relates the sine of three times an angle to the sine of the single angle. The formula is stated as:
\[ \sin 3x = 3\sin x - 4\sin^3 x \]
This identity is very useful for simplifying expressions involving powers of sine or for expanding expressions to a single sine term of a multiple angle. It can be derived by using the angle sum formula \(\sin(A+B) = \sin A \cos B + \cos A \sin B\) and applying it to \(\sin(2x + x)\), along with the double angle formulas for \(\sin 2x\) and \(\cos 2x\).
Expression 3sin x - 4sin3x Evaluation
Let's consider the given expression that needs to be evaluated:
\[ 3\sin x - 4\sin^3 x \]
When we compare this expression directly with the triple angle formula for sine, we observe an exact match:
- The given expression is: \(3\sin x - 4\sin^3 x\)
- The triple angle formula for sine is: \(\sin 3x = 3\sin x - 4\sin^3 x\)
Since the right-hand side of the triple angle formula is identical to the expression provided, we can conclude that the expression \(3\sin x - 4\sin^3 x\) evaluates to \(\sin 3x\).
Steps to Evaluate 3sinx - 4sin3x
Here’s a clear breakdown of the steps involved in evaluating the given expression:
- Step 1: Identify the structure of the expression.
The expression \(3\sin x - 4\sin^3 x\) involves \(\sin x\) and its cube, \(\sin^3 x\). This specific combination often points to a multiple-angle identity. - Step 2: Recall relevant trigonometric identities.
Consider the fundamental trigonometric identities, especially those dealing with double or triple angles. The triple angle formula for sine is a strong candidate because of the presence of \(\sin x\) and \(\sin^3 x\). - Step 3: Apply the triple angle formula for sine.
Remember the identity: \(\sin 3x = 3\sin x - 4\sin^3 x\). - Step 4: Substitute directly.
Since the expression \(3\sin x - 4\sin^3 x\) is precisely the right-hand side of the \(\sin 3x\) identity, we can substitute it with \(\sin 3x\).
Therefore, the evaluation of \(3\sin x - 4\sin^3 x\) is \(\sin 3x\).
Options Comparison
Let's compare our derived result with the given options to find the correct answer.
| Option Number | Option Text | Match with Result |
|---|---|---|
| 1 | \(2\sin 3x\) | No, our result is \(\sin 3x\), not \(2\sin 3x\). |
| 2 | \( \sin 3x \) | Yes, this matches our derived result exactly. |
| 3 | \( \cos 2x \cos x \) | No, this is a product of cosines and does not simplify to \(\sin 3x\). |
| 4 | \( \sin x \) | No, the expression involves a cubic term and simplifies to a triple angle, not a single angle. |
Based on our evaluation using the triple angle formula, the expression \(3\sin x - 4\sin^3 x\) simplifies to \(\sin 3x\).
Important Questions from Trigonometric Identities
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If \(\sin \left( {A - B} \right) = \frac{1}{2}\) and \(\cos \left( {A + B} \right) = \frac{1}{2}\) , where A > B > 0° and A + B is an acute angle, then the value of A is:
Find the value of $\cos 10^\circ \times \cos 30^\circ \times \cos 50^\circ \times \cos 70^\circ$
