subtended at the centre of a circle of radius cm by an arc of length cm (Use π ). . In a circle of diameter cm, the length of a chord is cm. Find the length of minor arc of the chord.
. If in two circles, arcs of the same length subtend angles ° and ° at the centre, find the ratio of their radii. . Find the angle in radian through which a pendulum swings if its length is cm and th e tip describes an arc of length cm cm (iii) cm .
Trigonometric Functions In earlier classes, we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions. Consider a unit circle with centre at origin of the coordinate axes. Let P ( a, b ) be any point on the circle with angle AOP = x radian, i.e., length of arc AP = x (Fig .
). We define cos x = a and sin x = b Since ∆ OMP is a right triangle, we have OM + MP = OP or a + b = Thus, for every point on the unit circle, we have a + b = or cos x + sin x = Since one complete revolution subtends an angle of π radian at the centre of the circle, ∠ AOB = π , Fig . MATHEMATICS ∠ AOC = π and ∠ AOD = π . All angles which are integral multiples of π are called quadrantal angles .
The coordinates of the points A, B, C and D are, respectively, ( , ), ( , ), (– , ) and ( , – ). Therefore, for quadrantal angles, we have cos ° = sin ° = , cos π = sin π