ABSTRACT
By convention, when OA moves anticlockwise angular measurement is considered positive, and vice-versa.
(ii) Let OA be rotated anticlockwise so that θ1 is any angle in the first quadrant and let perpendicular AB be constructed to form the right-angled triangle OAB (see Fig. 15.3). Since all three sides of the triangle are positive, all six trigonometric ratios are positive in the first quadrant. (Note: OA is always positive since it is the radius of a circle.)
(iii) Let OA be further rotated so that θ2 is any angle in the second quadrant and let AC be constructed to form the right-angled triangle
OAC. Then:
sin θ2 = ++ = + cos θ2 = − + = −
tan θ2 = +− = − cosec θ2 = + + = +
sec θ2 = +− = − cot θ2 = − + = −
(iv) Let OA be further rotated so that θ3 is any angle in the third quadrant and let AD be constructed to form the right-angled triangle OAD. Then:
sin θ3 = −+ = − (and hence cosec θ3 is −)
cos θ3 = −+ = − (and hence sec θ3 is +)
tan θ3 = −− = + (and hence cot θ3 is −) (v) Let OA be further rotated so that θ4 is any angle
in the fourth quadrant and let AE be constructed to form the right-angled triangle OAE. Then:
sin θ4 = −+ = − (and hence cosec θ4 is −)
cos θ4 = ++ = + (and hence sec θ4 is +)
tan θ4 = −+ = − (and hence cot θ4 is −)
(vi) The results obtained in (ii) to (v) are summarized in Fig. 15.4. The letters underlined spell the word CAST when starting in the fourth quadrant and moving in an anticlockwise direction.
