Basic Engineering Mathematics, Fifth Edition

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, angleθ=cos−^1 ( 0. 2348 )= 76. 42 ◦.

Measured from 0◦, the two angles whose cosine

is− 0 .2348 areα= 180 ◦− 76. 42 ◦,i.e. 103. 58 ◦and

α= 180 ◦+ 76. 42 ◦,i.e. 256. 42 ◦

Now try the following Practice Exercise

PracticeExercise 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 ofx
   between 0◦and 360◦.

(a) x=cos−^10. 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◦to360◦, giving the answers in degrees and
   minutes.


4. cos−^1 (− 0. 5316 )=t


5. sin−^1 (− 0. 6250 )=α


6. tan−^10. 8314 =θ

22.3 The production of sine and cosine waves

In Figure22.11, letORbe a vector 1 unitlongand free to

rotate anticlockwiseabout 0. In one revolutiona circle is

producedandisshownwith15◦sectors.Each radius arm

has a vertical and a horizontalcomponent. For example,

at 30◦, the vertical component isTSand the horizontal

component isOS.

From triangleOST,

sin30◦=

TS

TO

=

TS

1

i.e. TS=sin30◦

and cos30◦=

OS

TO

=

OS

1

i.e. OS=cos30◦

1208

908

608

3608

3308 2 0.5

2 1.0

1.0

T 0.5

y

R

S

S 9

T 9

y 5 sinx

Angle x 8

308 608 1208 2108 2708 3308

3008

2708

2408

2108

1808

1508

O

Figure 22.11