In the `Delta ABCA gt B.` If the measures of A and B satisfy the equation `3 sin x - 4 sin ^(3) x - k =0, 0 lt k lt 1.` Then the measure of C is

Options

A. `pi//3`
B. `pi//2`
C. `2pi//3`
D. `5pi//6`

Answer

To solve the problem, we need to find the measure of angle C in triangle ABC, given that angle A is greater than angle B and that the angles satisfy the equation \(3 \sin x - 4 \sin^3 x - k = 0\) where \(0 < k < 1\).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We know that in triangle ABC, the angles A, B, and C must satisfy the triangle angle sum property, which states that \(A + B + C = 180^\circ\) or in radians, \(A + B + C = \pi\).

2. **Given Equation**:
The angles A and B satisfy the equation:
\[ 3 \sin x - 4 \sin^3 x - k = 0 \]
We can rearrange this to:
\[ 3 \sin x - 4 \sin^3 x = k \]

3. **Using the Sine Identity**:
The expression \(3 \sin x - 4 \sin^3 x\) can be recognized as the sine of a triple angle:
\[ \sin(3x) = 3 \sin x - 4 \sin^3 x \]
Therefore, we have:
\[ \sin(3x) = k \]

4. **Finding the Range of x**:
Since \(0 < k < 1\), it implies that \(3x\) must be in the range where the sine function is positive. This means:
\[ 0 < 3x < \frac{\pi}{2} \quad \text{or} \quad \frac{\pi}{2} < 3x < \frac{3\pi}{2} \]
However, since \(x\) must also be an angle in triangle ABC, we will focus on the first case, which gives:
\[ 0 < x < \frac{\pi}{6} \]

5. **Relating Angles A and B**:
Given that angle A is greater than angle B, we can denote:
\[ A = x \quad \text{and} \quad B = y \]
where \(x > y\).

6. **Using the Sine Condition**:
Since \(A + B + C = \pi\), we have:
\[ C = \pi - (A + B) = \pi - (x + y) \]

7. **Finding C**:
From the sine condition, we know:
\[ \sin(3A) = k \quad \text{and} \quad \sin(3B) = k \]
Since \(A > B\), it follows that:
\[ 3A = \pi - 3B \]
This leads to:
\[ 3A + 3B = \pi \quad \Rightarrow \quad A + B = \frac{\pi}{3} \]
Therefore, substituting back into the equation for C:
\[ C = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Answer:
The measure of angle C is:
\[ C = \frac{2\pi}{3} \]