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34: A Trig Formula for 34: A Trig Formula for the Area of a Trianglethe Area of a Triangle
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Trigonometry
Module C2
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Trigonometry
In a right angled triangle, the 3 trig ratios for an angle x are defined as follows:
hypotenuse
oppositexsin
3 Trig Ratios: A reminder
opposite
hypotenuse
x
Trigonometry
In a right angled triangle, the 3 trig ratios for an angle x are defined as follows:
hypotenuse
adjacentxcos
hypotenuse
xadjacent
3 Trig Ratios: A reminder
Trigonometry
In a right angled triangle, the 3 trig ratios for an angle x are defined as follows:
adjacent
oppositextan
opposite
xadjacent
3 Trig Ratios: A reminder
Trigonometry
Using the trig ratios we can find unknown angles and sides of a right angled triangle, provided that, as well as the right angle, we know the following:
either 1 side and 1 angleor 2 sides
3 Trig Ratios: A reminder
Trigonometry
730
y
e.g. 1 y
730sin
30sin
7y
14y
8
10tan x
e.g. 2 10
8
x351x (3
s.f.)
Tip: Always start with the trig ratio, whether or not
you know the angle.
3 Trig Ratios: A reminder
Trigonometry
Scalene Triangles
We will now find a formula for the area of a triangle that is not right angled, using 2 sides and 1 angle.
Trigonometry
a, b and c are the sides opposite angles A, B and C respectively. ( This is a conventional way of labelling a triangle ).
ABC is a non-right angled triangle.
A B
C
b a
c
Area of a Triangle
Trigonometry
Draw the perpendicular, h, from C to BA.
N
h
height base Area 2
1
hc21 Area
C
b a
c A B
Area of a TriangleABC is a non-right angled triangle.
Trigonometry
Draw the perpendicular, h, from C to BA.
N
h
height base Area 2
1
hc21 Area - - - - - (1)
In ,ΔACN
C
b a
c A B
Area of a TriangleABC is a non-right angled triangle.
Trigonometry
Draw the perpendicular, h, from C to BA.
N
h
height base Area 2
1
hc21 Area - - - - - (1)
In ,ΔACNb
hA sin
C
b a
c A B
Area of a TriangleABC is a non-right angled triangle.
Trigonometry
hAb sin
h b a
c
c
C
N A B
height base Area 2
1
hc21 Area - - - - - (1)
In ,ΔACNb
hA sin
Draw the perpendicular, h, from C to BA.
Area of a TriangleABC is a non-right angled triangle.
Trigonometry
h b a
c
a
c
C
N B Substituting for h in (1)A
height base Area 2
1
hc21 Area - - - - - (1)
In ,ΔACNb
hA sin
Draw the perpendicular, h, from C to BA.
hAb sin
Area of a Triangle
c21 Area Ab sin
h
ABC is a non-right angled triangle.
Trigonometry
c
Abc sin21 Area
b a a
C
B A Substituting for h in (1)
height base Area 2
1
hc21 Area - - - - - (1)
In ,ΔACNb
hA sin
Draw the perpendicular, h, from C to BA.
hAb sin
Area of a Triangle
c21 Area Ab sin
ABC is a non-right angled triangle.
Trigonometry
Any side can be used as the base, so
Area of a Triangle
• The formula always uses 2 sides and the angle formed by those sides
Bca sin21Cab sin
21 Abc sin
21Area = = =
Trigonometry
Any side can be used as the base, so
Area of a Triangle
• The formula always uses 2 sides and the angle formed by those sides
c
b a a
C
B A
Bca sin21Cab sin
21 Abc sin
21Area = = =
Trigonometry
Any side can be used as the base, so
Area of a Triangle
• The formula always uses 2 sides and the angle formed by those sides
c
b a a
C
B A
Area = = = Bca sin21Cab sin
21 Abc sin
21
Trigonometry
Any side can be used as the base, so
Area of a Triangle
• The formula always uses 2 sides and the angle formed by those sides
c
b a a
C
B A
Area = = = Bca sin21Cab sin
21 Abc sin
21
Trigonometry
1. Find the area of the triangle PQR.
Example
7 cm
8 cm
R
Q P
80
36 64
Solution: We must use the angle formed by the 2 sides with the given lengths.
Trigonometry
1. Find the area of the triangle PQR.
Example
7 cm
8 cm
R
Q P
80
36 64
Solution: We must use the angle formed by the 2 sides with the given lengths.
We know PQ and RQ so use angle Q
Trigonometry
1. Find the area of the triangle PQR.
Example
7 cm
8 cm
R
Q P
80
36 64
Solution: We must use the angle formed by the 2 sides with the given lengths.
We know PQ and RQ so use angle Q
64sin)8()7(21 Area
225 cm2 (3 s.f.)
Trigonometry
A useful application of this formula occurs when we have a triangle formed by 2 radii and a chord of a circle.
Area of a Triangle
r
B
A
C
r Cba sin
21 Area
sinrr21 Area
sin2r21 Area
Trigonometry
The area of triangle ABC is given by
SUMMARY
sin2r21
The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by
where is the angle between the radii.
Abc sin21Cab sin
21 Bca sin
21or or
Trigonometry
1. Find the areas of the triangles shown in the diagrams.
Exercises
radius = 4 cm.,
122XOY angle
(a) (b)
X 12 cm
9 cm
B A 28
C 36
Y
O
(a) cm2 (3 s.f.) (b) cm2 (3 s.f.)548 786Ans:
Trigonometry
Trigonometry
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Trigonometry
Any side can be used as the base, so
Area of a Triangle
• The formula always uses 2 sides and the angle formed by those sides
c
b a a
C
B A
Area = = = Bca sin21Cab sin
21 Abc sin
21
Trigonometry
e.g. Find the area of the triangle PQR.
7 cm
8 cm
R
Q P
80
36 64
Solution: We must use the angle formed by the 2 sides with the given lengths.
We know PQ and RQ so use angle Q
64sin)8()7(21 Area
225 cm2 (3 s.f.)
Trigonometry
The area of triangle ABC is given by
SUMMARY
sin2r21
The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by
where is the angle between the radii.
Abc sin21Cab sin
21 Bca sin
21or or