Trigo the sine and cosine rule from BA101

1. BA101
ENGINEERING
MATHEMATICS 1
Solve Triangular
Problems using
Sine and
Cosine Rules

2. MENU

3. SINE RULE
∆ABC is made up of 6 element: 3 sides (a, b, c)
denoted by the small letter of the vertices.
A
The sine rules enables
us to calculate sides
and angles
In the some triangles
where there is not a
right angle.
c
B
b
a
The Sine Rule is used to solve any problems involving triangles when at
least either of the following is known:
a) two angles and a side
b) two sides and an angle opposite
a given side (a non-included angle)
C

4. SINE RULE
We write the sine rule so that the
unknown angle or side is on the left of
the equation
When finding an angle:
sin A
a
sin B
b
sin C
c
b
sin B
c
sin C
When finding a side:
a
sin A

5. SINE RULE
1. In triangle ABC, b = 3.6 cm, c = 4.2 cm and angle
C = 110˚. Find the size of angles A and B.
C
3 6
A
sin B
a
4 2
Now find out
angle A ?
B
sin C
b
110 
c
sin B
B
A
3 6 sin 110
4 2
53 7


16 3

6. SINE RULE
2. In triangle PQR, PQ = 23 cm, angle R = 42˚ and
angle P = 17˚. Find the size of side QR.
p
sin P
r
sin R
p
23 sin 17
42
p
10.0cm

7. COSINE RULE
Sometimes the sine rule is not enough to
help us solve for a non-right angled
triangle. For example:
C
a
14
300
B
18
A
In the triangle shown, we do not have enough
information to use the sine rule. That is, the sine rule
only provided the Following:
a
14
18
sin 30 0 sin B sin C

8. COSINE RULE
For this reason we derive another useful result, known as
the COSINE RULE. The Cosine Rule maybe used when:
a.
Three sides are given
b.
Two sides and an included angle are given.
a
a
b
c
c
The cosine Rule:
a2
b2
c2
=
=
=
B
b2 + c2 – 2bc cos A
a2 + c2 - 2ac cos B
a2 + b2 - 2ab cos C

9. COSINE RULE
1.
In Triangle ABC, length of a = 7.5cm, length of b = 6.4cm and an
length of c = 5.8cm. Find the angle of C.
C
First you need to identify
where’s A, B, and C.
then…
B
c2
c os C
C
b
a
a2
b2
c
A
2ab c osC
a2
b2
c2
2ab
7.5 2
6.4 2
5.8 2
2(7.5)(6.4)
c os
1
0.6622
48.53

10. COSINE RULE
2.
In Triangle ABC, length of a = 1.8m, length of b = 2.3m
and an angle of C = 46˚. Find the length of c.
C
First you need to identify
where’s a, b, and c.
then…
b
a
B
Using the cosine rule,
c
c2
a 2 b 2 2ab cosC
c2
1.82
A
c
1.8
2.32 2(1.8)(2.3) cos 46
2
1.667 m
2.3
2
2(1.8)(2.3) cos 46

11. Question 1
From the diagram,
find the length of b and an angle of C.
cm
cm
b
sin B
b
a
sin A
a
sin B
sin A
5.63
sin 103.89
sin 51.4
5.63
0.9707
0.7815
7cm