Higher Engineering Mathematics

116 GEOMETRY AND TRIGONOMETRY

(b) the value of∠QPR.

[(a) 27.20 cm each (b) 45◦]

3. A man cycles 24 km due south and then 20 km
   due east. Another man, starting at the same
   time as the first man, cycles 32 km due east
   and then 7 km due south. Find the distance
   between the two men. [20.81 km]


4. A ladder 3.5 m long is placed against a per-
   pendicular wall with its foot 1.0 m from the
   wall. How far up the wall (to the nearest centi-
   metre) does the ladder reach? If the foot of the
   ladder is now moved 30 cm further away from
   the wall, how far does the top of the ladder
   fall? [3.35 m, 10 cm]


5. Two ships leave a port at the same time. One
   travels due west at 18.4 km/h and the other
   due south at 27.6 km/h. Calculate how far
   apart the two ships are after 4 hours.
   [132.7 km]

12.3 Trigonometric ratios of acute

angles

(a) With reference to the right-angled triangle

shown in Fig. 12.4:

(i) sineθ=

opposite side

hypotenuse

i.e. sinθ=

b

c

(ii) cosineθ=

adjacent side

hypotenuse

i.e. cosθ=

a

c

(iii) tangentθ=

opposite side

adjacent side

i.e. tanθ=

b

a

(iv) secantθ=

hypotenuse

adjacent side

i.e. secθ=

c

a

(v) cosecantθ=

hypotenuse

opposite side

i.e. cosecθ=

c

b

(vi) cotangentθ=

adjacent side

opposite side

i.e. cotθ=

a

b

Figure 12.4

(b) From above,

(i)

sinθ

cosθ

=

b

c

a

c

=

b

a

=tanθ,

i.e. tanθ=

sinθ

cosθ

(ii)

cosθ

sinθ

=

a

c

b

c

=

a

b

=cotθ,

i.e. cotθ=

cosθ

sinθ

(iii) secθ=

1

cosθ

(iv) cosecθ=

1

sinθ

(Note ‘s’ and ‘c’ go together)

(v) cotθ=

1

tanθ

Secants, cosecants and cotangents are called the

reciprocal ratios.

Problem 3. If cosX=

9

41

determine the value

of the other five trigonometry ratios.

Fig. 12.5 shows a right-angled triangleXYZ.

Since cosX=

9

41

, thenXY=9 units and

XZ=41 units.

Using Pythagoras’ theorem: 41^2 = 92 +YZ^2 from

whichYZ=

√

(41^2 − 92 )=40 units.