198 Basic Engineering Mathematics
S
1808
2708
08
3608
908
T
A
C
Figure 22.10
Cosine is positive in the first and fourth quadrants and
thus negative in the second and third quadrants – see
Figure 22.10 or from Figure 22.1(b).
In Figure 22.10, angle θ = cos −1 (0.2348) = 76.42 ◦ .
Measured from 0 ◦ , the two angles whose cosine
is −0.2348 are α = 180 ◦ − 76.42 ◦ , i.e. 103.58
◦ and
α = 180 ◦ + 76.42 ◦ , i.e. 256.42 ◦
Now try the following Practice Exercise
Practice Exercise 87 Angles of any
magnitude (answers on page 349)
1. Determine all of the angles between 0 ◦ and
360 ◦ whose sine is
(a) 0.6792
(b) −0.1483
2. Solve the following equations for values of x
between 0 ◦ and 360 ◦ .
(a) x = cos −1 0.8739
(b) x = cos −1 (−0.5572)
3. Find the angles between 0 ◦ to 360 ◦ whose
tangent is
(a) 0.9728
(b) −2.3420
In problems 4 to 6, solve the given equations in the
range 0 ◦ to 360 ◦ , giving the answers in degrees and
minutes.
4. cos −1 (−0.5316) = t
5. sin −1 (−0.6250) = α
6. tan −1 0.8314 = θ
22.3 The production of sine and
cosine waves
In Figure 22.11, let OR be a vector 1 unit long and free to
rotate anticlockwise about 0. In one revolution a circle is
produced and is shown with 15 ◦ sectors. Each radius arm
has a vertical and a horizontal component. For example,
at 30 ◦ , the vertical component is TS and the horizontal
component is OS.
From triangle OST,
sin 30 ◦ =
TS
TO
=
TS
1
i.e. TS = sin 30 ◦
and
cos 30 ◦ =
OS
TO
=
OS
1
i.e. OS = cos 30 ◦
1208
908
608
3608
3308
20.5
21.0
1.0
0.5
T
y
R
S
S9
T 9
y 5 sin x
Angle x 8
308 608
1208
2108
2708
3308
3008
2708
2408
2108
1808
1508
O
Figure 22.11
S
1808
2708
08
3608
908
T
A
C
Figure 22.10
Cosine is positive in the first and fourth quadrants and
thus negative in the second and third quadrants – see
Figure 22.10 or from Figure 22.1(b).
In Figure 22.10, angle θ = cos −1 (0.2348) = 76.42 ◦ .
Measured from 0 ◦ , the two angles whose cosine
is −0.2348 are α = 180 ◦ − 76.42 ◦ , i.e. 103.58
◦ and
α = 180 ◦ + 76.42 ◦ , i.e. 256.42 ◦
Now try the following Practice Exercise
Practice Exercise 87 Angles of any
magnitude (answers on page 349)
1. Determine all of the angles between 0 ◦ and
360 ◦ whose sine is
(a) 0.6792
(b) −0.1483
2. Solve the following equations for values of x
between 0 ◦ and 360 ◦ .
(a) x = cos −1 0.8739
(b) x = cos −1 (−0.5572)
3. Find the angles between 0 ◦ to 360 ◦ whose
tangent is
(a) 0.9728
(b) −2.3420
In problems 4 to 6, solve the given equations in the
range 0 ◦ to 360 ◦ , giving the answers in degrees and
minutes.
4. cos −1 (−0.5316) = t
5. sin −1 (−0.6250) = α
6. tan −1 0.8314 = θ
22.3 The production of sine and
cosine waves
In Figure 22.11, let OR be a vector 1 unit long and free to
rotate anticlockwise about 0. In one revolution a circle is
produced and is shown with 15 ◦ sectors. Each radius arm
has a vertical and a horizontal component. For example,
at 30 ◦ , the vertical component is TS and the horizontal
component is OS.
From triangle OST,
sin 30 ◦ =
TS
TO
=
TS
1
i.e. TS = sin 30 ◦
and
cos 30 ◦ =
OS
TO
=
OS
1
i.e. OS = cos 30 ◦
1208
908
608
3608
3308
20.5
21.0
1.0
0.5
T
y
R
S
S9
T 9
y 5 sin x
Angle x 8
308 608
1208
2108
2708
3308
3008
2708
2408
2108
1808
1508
O
Figure 22.11