Higher Engineering Mathematics

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

1. Find all the angles between 0◦and 360◦
   whose sine is−0.7321.
   [227◦ 4 ′and 312◦ 56 ′]


2. Determine the angles between 0◦and 360◦
   whose cosecant is 2.5317.
   [23◦ 16 ′and 156◦ 44 ′]


3. 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.

4. cos−^1 (− 0 .5316)=t
   [t= 122 ◦ 7 ′and 237◦ 53 ′]


5. sec−^12. 3162 =x
   [x= 64 ◦ 25 ′and 295◦ 35 ′]


6. 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