The figures show the angles whose measures are radian, – radian, radian and – radian. (iii) Fig . (i) to (iv) (iv) We know that the circumference of a circle of radius unit is π . Thus, one complete revolution of the initial side subtends an angle of π radian.
More generally, in a circle of radius r , an arc of length r will subtend an angle of radian. It is well-known that equal arcs of a circle subtend equal angle at the centre. Since in a circle of radius r , an arc of length r subtends an angle whose measure is radian, an arc of length l will subtend an angle whose measure is l r radian. Thus, if in a circle of radius r , an arc of length l subtends an angle θ radian at the centre, we have θ = l r or l = r θ.
MATHEMATICS . . Relation between radian and real numbers Consider the unit circle with centre O. Let A be any point on the circle.
Consider OA as initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the centre of the circle. Consider the line PAQ which is tangent to the circle at A. Let the point A represent the real number zero, AP represents positive real number and AQ represents negative real numbers (Fig .
). If we rope the line AP in the anticlockwise direction along the circle, and AQ in the clockwise direction, then every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered as one and the same. .
. Relation between degree and radian Since a circle subtends at the centre an angle whose radian measure is π and its degree measure is °, it follows that π radian = ° or π radian = ° The above relation enables us to express