Multiple choice

What is the length of the chord of a unit circle which substends an angle $\theta$ at the centre ?

  1. $\sin(\dfrac{\theta}{2})$
  2. $\cos(\dfrac{\theta}{2})$
  3. $2\sin(\dfrac{\theta}{2})$
  4. $2\cos(\dfrac{\theta}{2})$
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C Correct answer
Explanation

In a unit circle, a chord subtending angle theta at the center forms an isosceles triangle with two radii of length 1. Using trigonometry, the length of the chord is 2 * r * sin(theta/2), where r=1.

AI explanation

Drawing a perpendicular from the center to the chord bisects both the chord and the central angle, creating a right triangle with hypotenuse 1 and angle theta/2. Using the definition of sine, sin(theta/2) = (half chord length)/1, so the half chord length is sin(theta/2). Multiplying by two gives the full chord length as 2sin(theta/2).

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