The distance between the points `(a cosalpha, a sinalpha)` and `(a cosbeta, a sinbeta)` where a> 0

Options

A. `2|asin ""(alpha-beta)/(2)|`
B. `|a sin ""(alpha-beta)/(2)|`
C. `2|a sin ""(alpha+beta)/(2)|`
D. `|asin""(alphs+beta)/(2)|`

Answer

To find the distance between the points \((a \cos \alpha, a \sin \alpha)\) and \((a \cos \beta, a \sin \beta)\), we will use the distance formula. Here’s the step-by-step solution:

### Step 1: Identify the Points
The points given are:
- Point 1: \((x_1, y_1) = (a \cos \alpha, a \sin \alpha)\)
- Point 2: \((x_2, y_2) = (a \cos \beta, a \sin \beta)\)

### Step 2: Use the Distance Formula
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 3: Substitute the Points into the Formula
Substituting the coordinates of the points into the distance formula:
\[ d = \sqrt{(a \cos \beta - a \cos \alpha)^2 + (a \sin \beta - a \sin \alpha)^2} \]

### Step 4: Factor Out \(a\)
Factor \(a\) from both terms inside the square root:
\[ d = \sqrt{a^2 \left((\cos \beta - \cos \alpha)^2 + (\sin \beta - \sin \alpha)^2\right)} \]
This simplifies to:
\[ d = a \sqrt{(\cos \beta - \cos \alpha)^2 + (\sin \beta - \sin \alpha)^2} \]

### Step 5: Expand the Terms
Now, we will expand the terms inside the square root:
\[ (\cos \beta - \cos \alpha)^2 = \cos^2 \beta + \cos^2 \alpha - 2 \cos \alpha \cos \beta \]
\[ (\sin \beta - \sin \alpha)^2 = \sin^2 \beta + \sin^2 \alpha - 2 \sin \alpha \sin \beta \]

### Step 6: Combine the Expanded Terms
Now combine the two expansions:
\[ d = a \sqrt{\left(\cos^2 \beta + \sin^2 \beta\right) + \left(\cos^2 \alpha + \sin^2 \alpha\right) - 2(\cos \alpha \cos \beta + \sin \alpha \sin \beta)} \]
Using the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\):
\[ d = a \sqrt{1 + 1 - 2(\cos \alpha \cos \beta + \sin \alpha \sin \beta)} \]
This simplifies to:
\[ d = a \sqrt{2 - 2(\cos \alpha \cos \beta + \sin \alpha \sin \beta)} \]

### Step 7: Use the Cosine of the Angle Difference
Recognizing the cosine of the angle difference:
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
Thus, we can rewrite the expression:
\[ d = a \sqrt{2 - 2\cos(\alpha - \beta)} \]

### Step 8: Factor Out the Common Terms
Factoring out the common terms gives:
\[ d = a \sqrt{2(1 - \cos(\alpha - \beta))} \]

### Step 9: Use the Identity for Sine
Using the identity \(1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right)\):
\[ d = a \sqrt{2 \cdot 2 \sin^2\left(\frac{\alpha - \beta}{2}\right)} \]
This simplifies to:
\[ d = 2a \left|\sin\left(\frac{\alpha - \beta}{2}\right)\right| \]

### Final Result
Thus, the distance between the two points is:
\[ d = 2a \left|\sin\left(\frac{\alpha - \beta}{2}\right)\right| \]