Distance between the points `A(acosalpha, "asin alpha)` and `B(acosbeta," asinbeta)` is equal to

Options

A. `2asin((alpha+beta)/(2))`
B. `2acos((alpha+beta)/(2))`
C. `2asin((alpha-beta)/(2))`
D. `2acos((alpha-beta)/(2))`

Answer

To find the distance between the points \( A(a \cos \alpha, a \sin \alpha) \) and \( B(a \cos \beta, a \sin \beta) \), we will use the distance formula in coordinate geometry. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B:**
- \( A = (a \cos \alpha, a \sin \alpha) \)
- \( B = (a \cos \beta, a \sin \beta) \)

2. **Substitute the coordinates into the distance formula:**
\[ d = \sqrt{(a \cos \beta - a \cos \alpha)^2 + (a \sin \beta - a \sin \alpha)^2} \]

3. **Factor out \( a \) from both terms:**
\[ d = \sqrt{a^2 \left( (\cos \beta - \cos \alpha)^2 + (\sin \beta - \sin \alpha)^2 \right)} \]
\[ d = a \sqrt{(\cos \beta - \cos \alpha)^2 + (\sin \beta - \sin \alpha)^2} \]

4. **Use trigonometric identities:**
- We can use the identities:
\[ \cos \beta - \cos \alpha = -2 \sin\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right) \]
\[ \sin \beta - \sin \alpha = 2 \cos\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right) \]

5. **Substitute these identities into the distance formula:**
\[ d = a \sqrt{\left(-2 \sin\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right)\right)^2 + \left(2 \cos\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right)\right)^2} \]

6. **Factor out common terms:**
\[ d = a \sqrt{4 \sin^2\left(\frac{\beta - \alpha}{2}\right) \left( \sin^2\left(\frac{\beta + \alpha}{2}\right) + \cos^2\left(\frac{\beta + \alpha}{2}\right) \right)} \]

7. **Use the Pythagorean identity:**
\[ \sin^2 x + \cos^2 x = 1 \]
Thus,
\[ d = a \sqrt{4 \sin^2\left(\frac{\beta - \alpha}{2}\right)} = a \cdot 2 \left| \sin\left(\frac{\beta - \alpha}{2}\right) \right| \]

8. **Final expression for distance:**
\[ d = 2a \left| \sin\left(\frac{\beta - \alpha}{2}\right) \right| \]

### Conclusion:
The distance between the points \( A(a \cos \alpha, a \sin \alpha) \) and \( B(a \cos \beta, a \sin \beta) \) is given by:
\[ d = 2a \left| \sin\left(\frac{\beta - \alpha}{2}\right) \right| \]