The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then `A^(4)` is equal to ______.

Answer

To solve the problem of finding the area \( A \) enclosed between the graphs of \( y = \sin x \) and \( y = \cos x \) between two consecutive points of intersection, we will follow these steps:

### Step 1: Identify Points of Intersection
The sine and cosine functions intersect when \( \sin x = \cos x \). This occurs at:
\[ x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
Thus, the first two consecutive points of intersection are \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

### Step 2: Set Up the Integral for Area
The area \( A \) between the curves from \( x = \frac{\pi}{4} \) to \( x = \frac{5\pi}{4} \) can be found using the integral:
\[ A = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} (\sin x - \cos x) \, dx \]
Here, \( \sin x \) is the upper curve and \( \cos x \) is the lower curve in this interval.

### Step 3: Compute the Integral
Now we compute the integral:
\[ A = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \sin x \, dx - \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos x \, dx \]
Calculating each integral separately:
1. The integral of \( \sin x \):
\[ \int \sin x \, dx = -\cos x \]
Evaluating from \( \frac{\pi}{4} \) to \( \frac{5\pi}{4} \):
\[ [-\cos x]_{\frac{\pi}{4}}^{\frac{5\pi}{4}} = -\cos\left(\frac{5\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right) = -\left(-\frac{1}{\sqrt{2}}\right) + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

2. The integral of \( \cos x \):
\[ \int \cos x \, dx = \sin x \]
Evaluating from \( \frac{\pi}{4} \) to \( \frac{5\pi}{4} \):
\[ [\sin x]_{\frac{\pi}{4}}^{\frac{5\pi}{4}} = \sin\left(\frac{5\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = -\sqrt{2} \]

### Step 4: Combine Results
Now we combine the results:
\[ A = \sqrt{2} - (-\sqrt{2}) = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]

### Step 5: Calculate \( A^4 \)
Finally, we need to find \( A^4 \):
\[ A^4 = (2\sqrt{2})^4 = 2^4 \cdot (\sqrt{2})^4 = 16 \cdot 4 = 64 \]

Thus, the value of \( A^4 \) is:
\[ \boxed{64} \]