Principle Of Mathematical Induction

Question
CBSEENMA11014282

Prove that the number of radians in an angle subtended by an arc of a circle at the centre   = space space fraction numerator arc space length over denominator radius end fraction

Solution

Draw a circle of radius OA = r with centre at O.
Cut off an arc AB of length r and join OB; then angle AOB space equals 1 radian.
Cut off an another arc ABC of length l and join OC such that angle AOC space equals space straight theta (radians). Since angles at the centre of the circle are proportional to the arcs on which they stand,
 ∴              fraction numerator angle AOC over denominator angle AOB end fraction equals fraction numerator arc space ABC over denominator arc space AB end fraction space rightwards double arrow space space theta over 1 equals l over r space space space space rightwards double arrow space space space theta space left parenthesis r a d i a n s right parenthesis space equals space l over r
                                                                

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