Express `sin "" (5pi)/(2)` in terms of ` cos 5pi ,`

Answer

To express \( \sin\left(\frac{5\pi}{2}\right) \) in terms of \( \cos(5\pi) \), we can follow these steps:

### Step 1: Simplify \( \sin\left(\frac{5\pi}{2}\right) \)
The angle \( \frac{5\pi}{2} \) can be simplified. Since the sine function has a period of \( 2\pi \), we can subtract \( 2\pi \) from \( \frac{5\pi}{2} \):

\[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \]

Thus, we have:

\[ \sin\left(\frac{5\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) \]

### Step 2: Evaluate \( \sin\left(\frac{\pi}{2}\right) \)
We know that:

\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 3: Find \( \cos(5\pi) \)
Next, we need to evaluate \( \cos(5\pi) \). Since the cosine function also has a period of \( 2\pi \):

\[ 5\pi - 2\pi \cdot 2 = 5\pi - 4\pi = \pi \]

Thus, we have:

\[ \cos(5\pi) = \cos(\pi) = -1 \]

### Step 4: Express \( \sin\left(\frac{5\pi}{2}\right) \) in terms of \( \cos(5\pi) \)
Now, we can express \( \sin\left(\frac{5\pi}{2}\right) \) in terms of \( \cos(5\pi) \):

Since \( \cos(5\pi) = -1 \), we can relate \( \sin\left(\frac{5\pi}{2}\right) \) to \( \cos(5\pi) \):

\[ \sin\left(\frac{5\pi}{2}\right) = 1 \]

Thus, we can express it as:

\[ \sin\left(\frac{5\pi}{2}\right) = -\frac{1}{2} \cdot \cos(5\pi) + \frac{3}{2} \]

This gives us a relationship between \( \sin\left(\frac{5\pi}{2}\right) \) and \( \cos(5\pi) \).

### Final Result:
\[ \sin\left(\frac{5\pi}{2}\right) = 1 \]