Trigonometric functions and their properties

. . Trigonometric Functions of any angle in terms of Cartesian coordinates P(x,y) r O θ Figure . We have studied the principles of trigonometric ratios in the lower classes using acute angles .

But we come across many angles which are not acute. We shall extend the acute angle idea and define trigonometric functions for any angle. The trigonometric ratios to any angle in terms of radian measure are called trigonometric functions. Let P ( x, y ) be a point other than the origin on the terminal side of an angle θ in standard position .

Let OP = r. Thus, r = p x + y The six trigonometric functions of θ are defined as follows: sin θ = y r and cos θ = x r . Using this, we have tan θ = y x , x = ; cot θ = x y , y = ; cosec θ = r y , y = ; sec θ = r x , x = . (i) Since | x | ≤ r, | y | ≤ r, we have | sin θ | ≤ and | cos θ | ≤ .

(ii) In the case of acute angle, the above definitions are equivalent to our earlier definitions using right triangle. (iii) The trigonometric functions have positive or negative values depending on the quadrant in which the point P ( x, y ) on the terminal side of θ lies. (iv) The above definitions of trigonometric functions is independent of the points on the terminal side of the angle. (verify!) Trigonometric ratios of Quadrantal angles θ