Higher Engineering Mathematics

150 GEOMETRY AND TRIGONOMETRY

(vii) In the first quadrant of Fig. 15.1 all the curves
have positive values; in the second only sine is
positive; in the third only tangent is positive;
in the fourth only cosine is positive (exactly as
summarized in Fig. 15.4).

A knowledge of angles of any magnitude is needed

when finding, for example, all the angles between

0 ◦and 360◦whose sine is, say, 0.3261. If 0.3261

is entered into a calculator and then the inverse

sine key pressed (or sin−^1 key) the answer 19.03◦

appears. However there is a second angle between

0 ◦and 360◦which the calculator does not give.

Sine is also positive in the second quadrant

(either from CAST or from Fig. 15.1(a)). The

other angle is shown in Fig. 15.5 as angle θ

whereθ= 180 ◦− 19. 03 ◦= 160. 97 ◦. Thus 19.03◦

and160.97◦are the angles between 0◦and 360◦

whose sine is 0.3261 (check that sin 160. 97 ◦=

0 .3261 on your calculator).

Figure 15.5

Be careful! Your calculator only gives you one

of these answers. The second answer needs to

be deduced from a knowledge of angles of any

magnitude, as shown in the following problems.

Problem 1. Determine all the angles between

0 ◦and 360◦whose sine is−0.4638.

The angles whose sine is−0.4638 occurs in the

third and fourth quadrants since sine is negative in

these quadrants (see Fig. 15.6(a)). From Fig. 15.6(b),

θ=sin−^10 .4638 = 27◦ 38 ′.

Measured from 0◦, the two angles between 0◦and

360 ◦whose sine is−0.4638 are 180◦+ 27 ◦ 38 ′, i.e.

207 ◦ 38 ′and 360◦− 27 ◦ 38 ′, i.e. 332 ◦ 22 ′. (Note that

a calculator generally only gives one answer, i.e.

− 27. 632588 ◦).

Figure 15.6

Problem 2. Determine all the angles between

0 ◦and 360◦whose tangent is 1.7629.

A tangent is positive in the first and third

quadrants (see Fig. 15.7(a)). From Fig. 15.7(b),

Figure 15.7