Cosine Ratio
Concept of Cosine Ratio
The cosine ratio of an acute angle  is defined as below:
adjacent side
cos θ 
hypotenuse
hypotenuse
θ
adjacent side of 
For a right-angled triangle with a given acute angle ,
the cosine ratio of  is a constant.
For example,
6
4
2
60
1
cos 60 =
60
60
2
2
1
3
=
=
4
2
6
3
In
ABC, C = 90, AB = 5.2 and AC = 2.
A
5.2
B
cos A 
AC
AB

2
5.2
5

13
2
C
 AC is the adjacent side of A,
and AB is the hypotenuse.
Follow-up question 5
In the following figures, find cos θ.
(a)
(b)
6
θ 3
5
8
θ
10
4
Solution
(a)
3
cos θ 
5
(b)
8
cos θ 
10
4

5
Example 8
In △PQR, ∠P  90, PQ  20, PR  21
and RQ  29. Find the values of
(a) cos ∠Q,
(b) cos ∠R.
(Give your answers in fractions.)
Solution
PQ
(a) cos Q 
QR
20

29
PR
(b) cos R 
QR
21

29
Finding Cosine Ratio Using Calculators
Find cos  for a given angle 
In degree mode, use the key cos to find the value of cos .
For example,
the value of cos 30 can be obtained by keying:
cos 30 EXE
The answer is 0.8660…
Follow-up question 6
By using a calculator, find the values of the following
expressions correct to 3 significant figures.
cos 40  cos 75
(a) 5 cos 29
(b)
2
Solution
(a) 5 cos 29
= 4.37
(b)
 cos 29 = 0.874 61…
(cor. to 3 sig. fig.)
cos 40  cos 75  cos 40 = 0.766 04…, cos 75 = 0.258 81…
2
= 0.512 (cor. to 3 sig. fig.)
Example 9
By using a calculator, find the values of the following
expressions correct to 4 decimal places.
7
cos 81
(b)
5
(a)
cos 12.3
(c)
cos 10
cos 72 
5
Solution
(a)
Keying sequence
Display
cos 12.3 EXE
0.977045574
cos 12.3  0.9770 (cor. to 4 d.p.)
(b)
Keying sequence
Display
( 7  5 ) cos 81 EXE
0.219008251
7
cos 81  0.2190 (cor. to 4 d.p.)
5
(c)
Keying sequence
Display
cos 72 – cos 10  5 EXE
0.112055443
cos 10
cos 72 
 0.1121 (cor. to 4 d.p.)
5
Find  from a given value of cos 
In degree mode, use the keys SHIFT and cos to
find the corresponding acute angle .
For example,
given that cos  = 0.5,  can be obtained by keying
SHIFT
cos 0.5 EXE
The answer is 60, i.e.  = 60.
Follow-up question 7
Find the acute angle  in each of the following using a
calculator. (Give your answers correct to 3 significant figures.)
(a) cos θ  0.474
cos 24
(b) cos θ 
2
Solution
(a) cos θ  0.474
θ  61.7
(cor. to 3 sig. fig.)
cos 24
 cos 24 = 0.913 54…
(b) cos θ 
2
θ  62.8 (cor. to 3 sig. fig.)
Example 10
Find the acute angles  in the following using a calculator.
(a) cos   0.583, correct to the nearest degree.
(b) cos   2 cos 75, correct to the nearest 0.1.
(c) 12 cos   5, correct to 3 significant figures.
Solution
(a)
Keying sequence
Display
SHIFT cos 0.583 EXE
54.33817552
cos   0.583
  54 (cor. to the nearest degree)
(b)
Keying sequence
Display
SHIFT cos ( 2  cos 75 ) EXE
58.8260478
cos   2 cos 75
  58.8 (cor. to the nearest 0.1)
(c) 12 cos   5
cos  
5
12
Keying sequence
Display
SHIFT cos ( 5  12 ) EXE
65.37568165
5
cos  
12
  65.4 (cor. to 3 sig. fig.)
Using Cosine Ratio to Find Unknowns in
Right-Angled Triangles
We can use the cosine ratio to solve problems involving
right-angled triangles.
In ABC, C = 90, B = 55 and AB = 8 m.
Find BC correct to 2 decimal places.
BC
8m
 cos B 
AB
BC
55
 cos 55  
B
8m
BC  8  cos 55 m
 4.59 m (cor. to 2 d.p.)
A
C
In PQR, R = 90, PQ = 9 m and QR = 7 m.
Find Q correct to 2 decimal places.
P
9m

QR
cos Q 
PQ

cos Q 
7m
9m
Q  38.94  (cor. to 2 d.p.)
Q
7m R
Follow-up question 8
In
ABC, C  90, AB  4 cm and BC  3.5 cm.
B
Find B correct to 2 decimal places.
3.5 cm
C
4 cm
Solution

BC
cos B 
AB

3.5 cm
cos B 
4 cm
B  28.96
A
(cor. to 2 d.p.)
Example 11
In △DEF, ∠D  90, ∠E = 62 and
EF = 8 cm. Find the length of DE
correct to 1 decimal place.
Example 12
In △PQR, ∠P  36, ∠Q 90 and
PQ  10 cm. Find the length of PR
correct to 1 decimal place.
Example 13
In △PQR, ∠R  90, PQ = 22 cm
and QR =18 cm. Find ∠Q correct to
the nearest 0.01.
Example 11
In △DEF, ∠D  90, ∠E = 62 and
EF = 8 cm. Find the length of DE
correct to 1 decimal place.
Solution
∵
∴
DE
cos E 
EF
DE
cos 62 
8 cm
DE  8 cos 62 cm
 3.8 cm (cor. to 1 d.p.)
Example 12
In △PQR, ∠P  36, ∠Q 90 and
PQ  10 cm. Find the length of PR
correct to 1 decimal place.
Solution
∵
∴
PQ
cos P 
PR
10 cm
cos 36 
PR
10
PR 
cm
cos 36
 12.4 cm (cor. to 1 d.p.)
Example 13
In △PQR, ∠R  90, PQ = 22 cm
and QR =18 cm. Find ∠Q correct to
the nearest 0.01.
Solution
∵
∴
QR
cos Q 
PQ
18 cm

22 cm
Q  35.10 (cor. to the nearest 0.01)