TRIGONOMETRIC WAVEFORMS 151
B
θ=tan−^11. 7629 = 60 ◦ 26 ′. Measured from 0◦, the
two angles between 0◦and 360◦whose tangent is
1.7629 are 60 ◦ 26 ′and 180◦+ 60 ◦ 26 ′, i.e. 240 ◦ 26 ′.
Problem 3. Solve sec−^1 (− 2 .1499)=α for
angles ofαbetween 0◦and 360◦.
Secant is negative in the second and third quad-
rants (i.e. the same as for cosine). From Fig. 15.8,
θ=sec−^12. 1499 =cos−^1
(
1
2. 1499
)
= 62 ◦ 17 ′.
Measured from 0◦, the two angles between 0◦and
360 ◦whose secant is−2.1499 are
α= 180 ◦− 62 ◦ 17 ′= 117 ◦ 43 ′ and
α= 180 ◦+ 62 ◦ 17 ′= 242 ◦ 17 ′
Figure 15.8
Problem 4. Solve cot−^11. 3111 =αfor angles
ofαbetween 0◦and 360◦.
Cotangent is positive in the first and third quad-
rants (i.e. same as for tangent). From Fig. 15.9,
θ=cot−^11. 3111 =tan−^1
(
1
1. 3111
)
= 37 ◦ 20 ′.
Figure 15.9
Henceα= 37 ◦ 20 ′
and α= 180 ◦+ 37 ◦ 20 ′= 217 ◦ 20 ′
Now try the following exercise.
Exercise 69 Further problems on evaluat-
ing trigonometric ratios of any magnitude
- Find all the angles between 0◦and 360◦
whose sine is−0.7321.
[227◦ 4 ′and 312◦ 56 ′] - Determine the angles between 0◦and 360◦
whose cosecant is 2.5317.
[23◦ 16 ′and 156◦ 44 ′] - If cotangentx=− 0 .6312, determine the val-
ues ofxin the range 0◦≤x≤ 360 ◦.
[122◦ 16 ′and 302◦ 16 ′]
In Problems 4 to 6 solve the given equations.
- cos−^1 (− 0 .5316)=t
[t= 122 ◦ 7 ′and 237◦ 53 ′] - sec−^12. 3162 =x
[x= 64 ◦ 25 ′and 295◦ 35 ′] - tan−^10. 8314 =θ
[θ= 39 ◦ 44 ′and 219◦ 44 ′]
15.3 The production of a sine and
cosine wave
In Figure 15.10, letORbe a vector 1 unit long
and free to rotate anticlockwise aboutO. 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 isTSand the horizontal compo-
nent isOS.
From trigonometric ratios,
sin 30◦=
TS
TO
=
TS
1
,i.e.TS=sin 30◦
and cos 30◦=
OS
TO
=
OS
1
,i.e.OS=cos 30◦
The vertical componentTSmay be projected across
toT′S′, which is the corresponding value of 30◦
on the graph ofyagainst anglex◦. If all such
vertical components asTSare projected on to the