https://file.aiquickdraw.com/mathbot/file/1730672567878te6y6pq7.jpg

Let's solve each part of this problem.

### Part (a): Show that \( \sin(3A) = 3 \sin A - 4 \sin^3 A \).

To show this identity, we will use the angle addition formulas. We know that:
\[
\sin(3A) = \sin(2A + A)
\]
Using the identity for \(\sin(x + y)\), we have:
\[
\sin(2A + A) = \sin(2A) \cos(A) + \cos(2A) \sin(A)
\]

Now, we use the double-angle identities:
1. \(\sin(2A) = 2 \sin(A) \cos(A)\),
2. \(\cos(2A) = 1 - 2 \sin^2(A)\).

Substitute these into the expression:
\[
\sin(3A) = (2 \sin A \cos A) \cos A + (1 - 2 \sin^2 A) \sin A
\]
Now simplify each term:
\[
= 2 \sin A \cos^2 A + \sin A - 2 \sin^3 A
\]

Since \(\cos^2 A = 1 - \sin^2 A\), replace \(\cos^2 A\):
\[
= 2 \sin A (1 - \sin^2 A) + \sin A - 2 \sin^3 A
\]
Expanding this expression:
\[
= 2 \sin A - 2 \sin^3 A + \sin A - 2 \sin^3 A
\]
Combine like terms:
\[
= 3 \sin A - 4 \sin^3 A
\]
Thus, we have shown that:
\[
\sin(3A) = 3 \sin A - 4 \sin^3 A
\]

### Part (b): Solve \( \sin(3x) = \sin(x) \) for \( 0 \leq x \leq 2\pi \).

To solve this equation, we can use the fact that if \(\sin(3x) = \sin(x)\), then \(3x\) and \(x\) must differ by integer multiples of \(2\pi\), or they must be supplementary angles. This gives two cases:

1. **Case 1:** \(3x = x + 2n\pi\), where \(n\) is an integer.
   - Solving for \(x\):
     \[
     2x = 2n\pi \Rightarrow x = n\pi
     \]
   - For \(0 \leq x \leq 2\pi\), the values of \(x\) are \(x = 0, \pi, 2\pi\).

2. **Case 2:** \(3x = \pi - x + 2n\pi\), where \(n\) is an integer.
   - Solving for \(x\):
     \[
     4x = \pi + 2n\pi \Rightarrow x = \frac{(2n+1)\pi}{4}
     \]
   - For \(0 \leq x \leq 2\pi\), this gives values:
     \[
     x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}
     \]

Combining all solutions, we have:
\[
x = 0, \pi, 2\pi, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}
\]

### Final Answer:
For part (b), the solutions are:
\[
x = 0, \frac{\pi}{4}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{7\pi}{4}, 2\pi
\]

Would you like further details on any step?

### Further Questions
1. How can you generalize the method used in part (a) for other trigonometric multiples?
2. What are some alternative ways to solve \( \sin(3x) = \sin(x) \)?
3. How can we use the unit circle to find solutions to \( \sin(3x) = \sin(x) \)?
4. Are there other trigonometric identities involving multiple angles that simplify similarly?
5. Can you use a graphical approach to confirm the solutions in part (b)?

### Tip
When solving trigonometric equations like \( \sin(3x) = \sin(x) \), consider both direct angle relationships and supplementary angle cases for a complete solution.

Ex. a) Show that sin(3A) = 3sin A - 4sin^3 A.

b) Solve sin(3x) = sin x, 0 ≤ x ≤ 2π.

Step-by-step solution for proving the trigonometric identity sin(3A) = 3sin A - 4sin^3 A and solving the equation sin(3x) = sin x for 0 ≤ x ≤ 2π. This problem covers essential trigonometric concepts and formulas, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation