trigonometry

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 perpendicular
wall with its foot 1.0 m from the
wall. How far up the wall (to the nearest centimetre)
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:
opposite side
(i) sine θ =
hypotenuse
(ii)
(iii)
(iv)
i.e.
i.e.
i.e.
sin θ = b c
adjacent side
cosine θ =
hypotenuse
cos θ = a c
opposite side
tangent θ =
adjacent side
tan θ = b a
secant θ = hypotenuse
adjacent side
(vi)
i.e.
Figure 12.4
(b) From above,
cotangent θ =
cot θ = a b
adjacent side
opposite side
b
sin θ
(i)
cos θ = c
a = b = tan θ,
a
c
i.e. tan θ = sin θ
cos θ
a
cos θ
(ii)
sin θ = c
= a = cot θ,
b b
c
i.e. cot θ = cos θ
sin θ
(iii) sec θ = 1
cos θ
(iv) cosec θ = 1
sin θ
(Note ‘s’ and ‘c’ go together)
(v) cot θ = 1
tan θ
Secants, cosecants and cotangents are called the
reciprocal ratios.
Problem 3. If cos X = 9 determine the value
41
of the other five trigonometry ratios.
(v)
i.e. sec θ = c a
cosecant θ = hypotenuse
opposite side
i.e.
cosec θ = c b
Fig. 12.5 shows a right-angled triangle XYZ.
Since cos X = 9 , then XY = 9 units and
41
XZ = 41 units.
Using Pythagoras’ theorem: 41 2 = 9 2 + YZ 2 from
which YZ = √ (41 2 − 9 2 ) = 40 units.