Basic Engineering Mathematics, Fifth Edition

196 Basic Engineering Mathematics

22.2 Angles of any magnitude

Figure 22.2 shows rectangular axesXX′ and YY′

intersecting at origin 0. As with graphical work, mea-

surements made to the right and above 0 are positive,

while those to the left and downwards are negative.

Let 0Abe free to rotate about 0. By convention,

when 0Amoves anticlockwise angular measurement is

considered positive, and vice versa.

3608

2708

1808

908

X 9 X

Y 9

Y

08

A

Quadrant 2

Quadrant 3 Quadrant 4

Quadrant 1

0

21

1

2

1

2

Figure 22.2

Let 0Abe rotated anticlockwise so thatθ 1 is any

angle in the first quadrant and let perpendicularAB

be constructed to form the right-angled triangle 0AB

in Figure 22.3. Since all three sides of the trian-

gle are positive, the trigonometric ratios sine, cosine

and tangent will all be positive in the first quadrant.

(Note: 0Ais always positive since it is the radius of a

circle.)

Let 0Abe further rotated so thatθ 2 is any angle in the

second quadrant and letACbe constructed to form the

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 22.3

right-angled triangle 0AC. Then,

sinθ 2 =

+

+

=+ cosθ 2 =

−

+

=−

tanθ 2 =

+

−

=−

Let 0Abe further rotated so thatθ 3 is any angle in the

third quadrant and letADbe constructed to form the

right-angled triangle 0AD. Then,

sinθ 3 =

−

+

=− cosθ 3 =

−

+

=−

tanθ 3 =

−

−

=+

Let 0Abe further rotated so thatθ 4 is any angle in the

fourth quadrant and letAEbe constructed to form the

right-angled triangle 0AE. Then,

sinθ 4 =

−

+

=− cosθ 4 =

+

+

=+

tanθ 4 =

−

+

=−

The above results are summarized in Figure 22.4, in

which all three trigonometric ratios are positive in the

first quadrant, only sine is positive in the second quad-

rant, only tangent is positive in the third quadrant and

only cosine is positive in the fourth quadrant.

The underlined letters in Figure 22.4 spell the word

CAST when starting in the fourth quadrant and moving

in an anticlockwise direction.

908

1808

2708

3608

08

Sine

Tangent Cosine

All positive

Figure 22.4

It is seen that, in the first quadrant of Figure 22.1,

all of the curves have positive values; in the second only

sine is positive; in the third only tangent is positive;

and in the fourth only cosine is positive – exactly as

summarized in Figure 22.4.