Question

Distance between two points \((a \cos \alpha, a \sin \alpha)\) and \((a \cos \beta, a \sin \beta)\) is equal to

• A. \(2a \sin \left(\frac{\alpha + \beta}{2}\right)\)
• B. \(2a \cos \left(\frac{\alpha + \beta}{2}\right)\)
• C. \(2a \sin \left(\frac{\alpha - \beta}{2}\right)\)
• D. \(2a \cos \left(\frac{\alpha - \beta}{2}\right)\)

Hint:

When finding the distance between two points in polar coordinates, use the trigonometric identities for sine and cosine differences.

Answer 1

The Correct Option is C

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the given points: \[ \text{Distance} = \sqrt{a^2 (\cos \beta - \cos \alpha)^2 + a^2 (\sin \beta - \sin \alpha)^2} \] Using the identity for the difference of cosines and sines: \[ \text{Distance} = 2a \sin \left(\frac{\alpha - \beta}{2}\right) \] Thus, the correct answer is \(2a \sin \left(\frac{\alpha - \beta}{2}\right)\).