Higher Engineering Mathematics, Sixth Edition

Trigonometric waveforms 135

curve appears to be discontinuous and repeats at

intervals of 180◦.

14.2 Angles of any magnitude

(i) Figure 14.2 shows rectangular axesXX’andYY’

intersecting at origin 0. As with graphical work,

measurements made to the right and above 0 are

positive while those to the left and downwards

are negative. LetOAbe free to rotate about 0.

By convention, whenOAmoves anticlockwise

angular measurement is considered positive, and

vice-versa.

908

3608

2708

1808

X 9 X

Y 9

Y

08

A

Quadrant 2

Quadrant 3 Quadrant 4

Quadrant 1

0

21

1

2

1

2

Figure 14.2

(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. 14.3). Since all three sides

of the triangle are positive, all six trigonometric

ratios are positivein the first quadrant. (Note:OA

is always positive since it is the radius of a circle.)

1808

908

2708

3608

 2 08

 3  4

 1

Quadrant 2

Quadrant 3 Quadrant 4

Quadrant 1

0

A

B

A A

C

D

E

A

2

2 2

1

1

1

1

1

11

Figure 14.3

(iii) LetOAbe further rotated so thatθ 2 is any angle

in the second quadrant and letACbe constructed

to form the right-angled triangleOAC. 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 summarized

inFig. 14.4. The letters underlinedspell the word

CAST when starting in the fourth quadrant and

moving in an anticlockwise direction.

908

1808

2708

3608

08

Sine (and cosecant)

positive

Tangent

(and cotangent)

positive

Cosine

(and secant)

positive

All positive

Figure 14.4