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