Basic Engineering Mathematics, Fifth Edition

Trigonometric waveforms 197

A knowledge of angles of any magnitude is needed
when finding, for example, all the angles between 0◦
and 360◦whosesineis,say,0.3261. If 0.3261 is entered
into acalculator and thentheinversesinekey pressed (or
sin−^1 key) the answer 19. 03 ◦appears. However, there
is a second angle between 0◦and 360◦which the cal-
culator does not give. Sine is also positive in the second
quadrant (either from CAST or from Figure 22.1(a)).
The other angle is shown in Figure 22.5 as angleθ,
where θ= 180 ◦− 19. 03 ◦= 160. 97 ◦.Thus,19. 03 ◦
and 160. 97 ◦are the angles between 0◦and 360◦whose
sine is 0.3261 (check that sin160. 97 ◦= 0 .3261 on your
calculator).

1808 19.03^8 19.03^8

2708

3608

08



908

SA

TC

Figure 22.5

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 worked problems.

Problem 1. Determine all of the angles between

0 ◦and 360◦whose sine is− 0. 4638

The angles whose sine is− 0 .4638 occur in the third
and fourth quadrants since sine is negative in these
quadrants – see Figure 22.6.

1.0

2 1.0

2 0.4638

(^0908180827083608)
207.63 8 332.37 8
x
y y 5 sinx
Figure 22.6
From Figure 22.7,θ=sin−^10. 4638 = 27. 63 ◦. Mea-
sured from 0◦, the two angles between 0◦and 360◦
whose sine is− 0 .4638 are 180◦+ 27. 63 ◦i.e. 207. 63 ◦
and 360◦–27. 63 ◦,i.e. 332. 37 ◦. (Note that if a calcu-
lator is used to determine sin−^1 (− 0. 4638 )it only gives
one answer:− 27. 632588 ◦.)
T
S A
C
908
1808
2708
3608
08
 
Figure 22.7
Problem 2. Determine all of the angles between
0 ◦and 360◦whose tangent is 1. 7629
A tangent is positive in the first and third quadrants –
see Figure 22.8.
1.7629
60.44 (^8) 240.44 8
0 908 1808 2708 3608
y 5 tan x
y
x
Figure 22.8
From Figure 22.9, θ=tan−^11. 7629 = 60. 44 ◦. Mea-
sured from 0◦, the two angles between 0◦and 360◦
whose tangent is 1.7629 are 60. 44 ◦and 180◦+ 60. 44 ◦,
i.e. 240. 44 ◦
1808
2708
3608
908
08
T C
S A


Figure 22.9
Problem 3. Solve the equation
cos−^1 (− 0. 2348 )=αfor angles ofαbetween 0◦
and 360◦