Higher Engineering Mathematics

INTRODUCTION TO TRIGONOMETRY 121

B

Hence 129. 9 +x=

75

tan 20◦

=

75

0. 3640

= 206 .0m

from which x= 206. 0 − 129. 9 = 76 .1m

Thus the ship sails 76.1 m in 1 minute, i.e. 60 s, hence
speed of ship

=

distance

time

=

76. 1

60

m/s

=

76. 1 × 60 × 60

60 × 1000

km/h=4.57 km/h

Now try the following exercise.

Exercise 55 Further problems on angles of

elevation and depression

1. If the angle of elevation of the top of a vertical
   30 m high aerial is 32◦,howfarisittothe
   aerial? [48 m]


2. From the top of a vertical cliff 80.0 m high
   the angles of depression of two buoys lying
   due west of the cliff are 23◦and 15◦, respec-
   tively. How far are the buoys apart?
   [110.1 m]


3. From a point on horizontal ground a surveyor
   measures the angle of elevation of the top of
   a flagpole as 18◦ 40 ′. He moves 50 m nearer
   to the flagpole and measures the angle of ele-
   vation as 26◦ 22 ′. Determine the height of the
   flagpole. [53.0 m]


4. A flagpole stands on the edge of the top of a
   building. At a point 200 m from the building
   the angles of elevation of the top and bot-
   tom of the pole are 32◦and 30◦respectively.
   Calculate the height of the flagpole.
   [9.50 m]


5. From a ship at sea, the angles of elevation of
   the top and bottom of a vertical lighthouse
   standing on the edge of a vertical cliff are
   31 ◦and 26◦, respectively. If the lighthouse is
   25.0 m high, calculate the height of the cliff.
   [107.8 m]


6. From a window 4.2 m above horizontal
   ground the angle of depression of the foot
   of a building across the road is 24◦and the
   angle of elevation of the top of the building is
   34 ◦. Determine, correct to the nearest centi-
   metre, the width of the road and the height of
   the building. [9.43 m, 10.56 m]
   7. The elevation of a tower from two points, one
   due east of the tower and the other due west of
   it are 20◦and 24◦, respectively, and the two
   points of observation are 300 m apart. Find
   the height of the tower to the nearest metre.
   [60 m]

12.6 Evaluating trigonometric ratios

Four-figure tables are available which gives sines,

cosines, and tangents, for angles between 0◦and

90 ◦. However, the easiest method of evaluating

trigonometric functions of any angle is by using a

calculator.

The following values, correct to 4 decimal places,

may be checked:

sine 18◦= 0 .3090, cosine 56◦= 0. 5592

sine 172◦= 0. 1392 cosine 115◦=− 0 .4226,

sine 241. 63 ◦=− 0 .8799, cosine 331. 78 ◦= 0. 8811

tangent 29◦= 0 .5543,

tangent 178◦=− 0. 0349

tangent 296. 42 ◦=− 2. 0127

To evaluate, say, sine 42◦ 23 ′using a calculator means

finding sine 42

23 ◦

60

since there are 60 minutes in

1 degree.

23

60

= 0. 383 ̇3 thus 42◦ 23 ′= 42. 383 ̇◦

Thus sine 42◦ 23 ′=sine 42. 383 ̇◦= 0 .6741, correct

to 4 decimal places.

Similarly, cosine 72◦ 38 ′=cosine 72

38 ◦

60

=0.2985,

correct to 4 decimal places.

Most calculators contain only sine, cosine and tan-

gent functions. Thus to evaluate secants, cosecants

and cotangents, reciprocals need to be used. The fol-

lowing values, correct to 4 decimal places, may be

checked:

secant 32◦=

1

cos 32◦

= 1. 1792

cosecant 75◦=

1

sin 75◦

= 1. 0353

cotangent 41◦=

1

tan 41◦

= 1. 1504

secant 215. 12 ◦=

1

cos 215. 12 ◦

=− 1. 2226