Principle Of Mathematical Induction

Question
CBSEENMA11014280

Prove that the radian is a constant angle.

Solution

Proof: Draw a circle  with centre at O and radius OA = r. Cut off an arc AB of length r. Join OA, OB and produce AO to meet the circle again in C.

Since the angles at the centre of a circle are proportional to the lengths of the arcs which subtend them.
 ∴  fraction numerator angle AOB over denominator angle AOC end fraction space equals space fraction numerator arc space AB over denominator arc space ABC end fraction


∴    fraction numerator 1 space radian over denominator 2 space right space angles end fraction space equals space fraction numerator straight r over denominator begin display style 1 half end style left parenthesis 2 πr right parenthesis end fraction            space space space space open square brackets because space arc space ABC space equals space 1 half space circumference close square brackets
 
rightwards double arrow             space space 1 radian equals fraction numerator 2 space right space angles over denominator straight pi end fraction equals constant

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Some More Questions From Principle of Mathematical Induction Chapter

Find 1 radian angle in degrees, minutes, seconds.