The angle between the curves `y=sin x and y = cos x, 0 lt x lt (pi)/(2)`, is

Options

A. `tan^(-1)(2sqrt(2))`
B. `tan^(-1)(3sqrt(2))`
C. `tan^(-1)(3sqrt(3))`
D. `tan^(-1)(5sqrt(2))`

Answer

To find the angle between the curves \( y = \sin x \) and \( y = \cos x \) in the interval \( 0 < x < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Find the intersection point of the curves
The curves intersect where \( \sin x = \cos x \).

Setting the two equations equal to each other:
\[ \sin x = \cos x \]

This can be rewritten as:
\[ \tan x = 1 \]

The solution to this equation in the interval \( 0 < x < \frac{\pi}{2} \) is:
\[ x = \frac{\pi}{4} \]

Now, we can find the corresponding \( y \)-coordinate:
\[ y = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Thus, the point of intersection is:
\[ \left(\frac{\pi}{4}, \frac{1}{\sqrt{2}}\right) \]

### Step 2: Calculate the slopes of the tangents at the intersection point
We need to find the derivatives of both functions.

For \( y_1 = \sin x \):
\[ \frac{dy_1}{dx} = \cos x \]
At \( x = \frac{\pi}{4} \):
\[ m_1 = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

For \( y_2 = \cos x \):
\[ \frac{dy_2}{dx} = -\sin x \]
At \( x = \frac{\pi}{4} \):
\[ m_2 = -\sin\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

### Step 3: Use the formula for the angle between two curves
The formula for the tangent of the angle \( \theta \) between two curves with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

Substituting the values we found:
\[ \tan \theta = \frac{\frac{1}{\sqrt{2}} - \left(-\frac{1}{\sqrt{2}}\right)}{1 + \left(\frac{1}{\sqrt{2}}\right)\left(-\frac{1}{\sqrt{2}}\right)} \]

This simplifies to:
\[ \tan \theta = \frac{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{1 - \frac{1}{2}} = \frac{\frac{2}{\sqrt{2}}}{\frac{1}{2}} = \frac{2\sqrt{2}}{1} = 2\sqrt{2} \]

### Step 4: Find the angle \( \theta \)
To find \( \theta \), take the arctangent:
\[ \theta = \tan^{-1}(2\sqrt{2}) \]

### Final Answer
Thus, the angle between the curves \( y = \sin x \) and \( y = \cos x \) in the interval \( 0 < x < \frac{\pi}{2} \) is:
\[ \theta = \tan^{-1}(2\sqrt{2}) \]