136 Higher Engineering Mathematics
(vii) In the first quadrant of Fig. 14.1 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. 14.4).
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. 14.1(a)).
The other angle is shown in Fig. 14.5 as angle θ
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 sin 160.97 ◦ = 0.3261 on your
calculator).
19.038
19.038
1808
2708
3608
08
908
S
A
T
C
Figure 14.5
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. 14.6(a)). From Fig. 14.6(b),
θ = sin −1 0.4638 = 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 ◦ ).
T
S
A
C
908
1808
2708
3608
08
1.0
21.0
20.4638
0
908 1808
2708
3608
3328429
2078389
x
y 5 sin x
y
(a)
(b)
Figure 14.6
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. 14.7(a)). From Fig. 14.7(b),
θ = tan −1 1.7629 =60 ◦ 26 . Measured from 0 ◦ , the two
1.7629
608269
2408269
0
3608
2708
1808
908
y 5 tan x
y
x
(a)
1808
2708
3608
908
08
C
T
S
A
(b)
Figure 14.7
(vii) In the first quadrant of Fig. 14.1 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. 14.4).
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. 14.1(a)).
The other angle is shown in Fig. 14.5 as angle θ
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 sin 160.97 ◦ = 0.3261 on your
calculator).
19.038
19.038
1808
2708
3608
08
908
S
A
T
C
Figure 14.5
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. 14.6(a)). From Fig. 14.6(b),
θ = sin −1 0.4638 = 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 ◦ ).
T
S
A
C
908
1808
2708
3608
08
1.0
21.0
20.4638
0
908 1808
2708
3608
3328429
2078389
x
y 5 sin x
y
(a)
(b)
Figure 14.6
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. 14.7(a)). From Fig. 14.7(b),
θ = tan −1 1.7629 =60 ◦ 26 . Measured from 0 ◦ , the two
1.7629
608269
2408269
0
3608
2708
1808
908
y 5 tan x
y
x
(a)
1808
2708
3608
908
08
C
T
S
A
(b)
Figure 14.7