Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 149

B

90 °

360 °

270 °

0

0 °

180 °

Quadrant 2 Quadrant 1

Y

Quadrant 3 Quadrant 4

X AX

Y







 

Figure 15.2

By convention, whenOAmoves anticlockwise

angular measurement is considered positive,

and vice-versa.

(ii) LetOAbe rotated anticlockwise so thatθ 1 is any

angle in the first quadrant and let perpendicular

ABbe constructed to form the right-angled tri-

angleOAB(see Fig. 15.3). Since all three sides

of the triangle are positive, all six trigonometric

ratios are positive in the first quadrant. (Note:

OAis always positive since it is the radius of a

circle.)

Figure 15.3

(iii) LetOAbe further rotated so thatθ 2 is any
angle in the second quadrant and letACbe
constructed to form the right-angled triangle

OAC. Then:

sinθ 2 =

+

+

=+ cosθ 2 =

−

+

=−

tanθ 2 =

+

−

=− cosecθ 2 =

+

+

=+

secθ 2 =

+

−

=− cotθ 2 =

−

+

=−

(iv) LetOAbe further rotated so thatθ 3 is any angle

in the third quadrant and letADbe constructed

to form the right-angled triangleOAD. Then:

sinθ 3 =

−

+

=−(and hence cosecθ 3 is−)

cosθ 3 =

−

+

=−(and hence secθ 3 is+)

tanθ 3 =

−

−

=+(and hence cotθ 3 is−)

(v) LetOAbe further rotated so thatθ 4 is any angle

in the fourth quadrant and letAEbe constructed

to form the right-angled triangleOAE. Then:

sinθ 4 =

−

+

=−(and hence cosecθ 4 is−)

cosθ 4 =

+

+

=+(and hence secθ 4 is+)

tanθ 4 =

−

+

=−(and hence cotθ 4 is−)

(vi) The results obtained in (ii) to (v) are sum-

marized in Fig. 15.4. The letters underlined

spell the word CAST when starting in the

fourth quadrant and moving in an anticlockwise

direction.

Figure 15.4