34 a trig formula for area of a triangle

7/25/2019 34 a Trig Formula for Area of a Triangle

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34: A Trig Formula for the34: A Trig Formula for the

Area of a TriangleArea of a Triangle

Christine Crisp

Teach A Level MathsTeach A Level Maths

Vol !: A" Core Mo#ulesVol !: A" Core Mo#ules


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Trigonometr$

Mo#ule C%

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Trigonometr$

&n a right angle#triangle' the 3 trig ratios for anangle xare #efine# as follo(s:

h$potenuse

oppositexsin

3 Trig )atios: A remin#er

oppositeh$potenuse

x


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Trigonometr$


h$potenuse

a#*acentxcos

h$potenuse

x

a#*acent



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Trigonometr$


a#*acent

oppositextan

opposite

x

a#*acent



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Trigonometr$

+sing the trig ratios (e can fin# un,no(n angles an#si#es of a right angle# triangle' provi#e# that' as(ell as the right angle' (e ,no( the follo(ing:

either! si#e an# ! angle

or% si#es



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Trigonometr$

7

30

y

eg ! y7

30sin

30sin

7y

14y

810tan x

eg %

10

8

x

351

x -3 sf.

Tip: Always start with the

trig ratio, whether or not

you know the angle.



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Trigonometr$

"calene Triangles

/e (ill no( fin# a formula for the area of atriangle that is not right angle#' using % si#es an#! angle


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Trigonometr$

a' ban# care the si#es opposite angles A' Ban# Crespectivel$ - This is a conventional(a$ of la0elling a triangle .

A1C is a non2right angle# triangle

A B

C

b a

c

Area of a Triangle


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Trigonometr$

ra( the perpen#icular' h' from Cto BA

N

h

height0aseArea 2

1

hc 21Area

C

b a

cA B

Area of a Triangle



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Trigonometr$


N

h

height0aseArea 2

1

hc 21Area 2 2 2 2 2 (1)

&n ,4AC5

C

b a

cA B

Area of a Triangle



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Trigonometr$


N

h

height0aseArea 2

1

hc 21Area 2 2 2 2 2 (1)

&n ,4AC5b

hAsin

C

b a

cA B

Area of a Triangle



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Trigonometr$

hAb sin

hb a

cc

C

NA B

height0aseArea

2

1

hc 21Area 2 2 2 2 2 (1)

&n ,4AC5b

hAsin


Area of a Triangle



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Trigonometr$

hb a

c

a

c

C

N B"u0stituting for hin (1) A

height0aseArea

2

1

hc 21Area 2 2 2 2 2 (1)

&n ,4AC5b

hAsin


hAb sin

Area of a Triangle

c%!Area Ab sin

h



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Trigonometr$

c

Abc sin2

1

Area

b aa

C

BA"u0stituting for hin (1)

height0aseArea

2

1

hc 21Area 2 2 2 2 2 (1)

&n ,4AC5b

hAsin


hAb sin

Area of a Triangle

c%!Area Ab sin



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Trigonometr$

An$ si#e can 0e use# as the 0ase' so

Area of a Triangle

The formula al(a$s uses %si#es an# the angle forme#0$ those si#es

Bca sin21Cab sin

21 Abc sin

21Area = = =


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Trigonometr$


Area of a Triangle


c

baa

C

BA

Bca sin21Cab sin

21 Abc sin

21Area = = =

T i


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Trigonometr$


Area of a Triangle


c

baa

C

BA

Area = = = Bca sin21Cab sin

21 Abc sin

21

T i


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Trigonometr$


Area of a Triangle


c

baa

C

BA


21 Abc sin

21

T i t


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Trigonometr$

! Fin# the area of the triangle 67)

89ample

cm

; cm

R

QP

80

36

64

"olution: /e must usethe angle forme# 0$ the% si#es (ith the givenlengths

T i t


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Trigonometr$


89ample

cm

; cm

R

QP

80

36

64


/e ,no( 67 an# )7 so use angle 7

T i t


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Trigonometr$


89ample

cm

; cm

R

QP

80

36

64



64sin)8()7(

%!Area

%%< cm% -3 sf.

T i t


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Trigonometr$

A useful application of this formula occurs (hen (ehave a triangle forme# 0$ % ra#ii an# a chor# of acircle

Area of a Triangle

r

B

A

C

rCba sin%!Area

sinrr%!Area

sin2

r%!Area

T i t


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Trigonometr$

The area of triangle A1C is given 0$

"+MMA)=

sin2r

%

!

The area of a triangle forme# 0$ % ra#ii oflength rof a circle an# the chor# *oining themis given 0$

(here is the angle 0et(een the ra#ii

Abc sin21Cab sin

21 Bca sin

21or or

T i n m t


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Trigonometr$

! Fin# the areas of the triangles sho(n in the#iagrams

89ercises

ra#ius = 4cm'

122XOYangle

-a. -0.

X!%cm

>cm

BA

28

C

36

Y

O

-a. cm%-3 sf. -0. cm%-3 sf.548 786

Ans:

Trigonometr$


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Trigonometr$

Trigonometr$


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Trigonometr$

The follo(ing sli#es contain repeats ofinformation on earlier sli#es' sho(n (ithoutcolour' so that the$ can 0e printe# an#

photocopie#

For most purposes the sli#es can 0e printe#as ?an#outs (ith up to 6sli#es per sheet


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Trigonometr$


Area of a Triangle


c

baa

C

BA


21 Abc sin

21


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Trigonometr$

eg Fin# the area of the triangle 67)

cm

; cm

R

QP

80

36

64



64sin)8()7(

%!Area

%%< cm% -3 sf.


30/30

Trigonometr$

The area of triangle A1C is given 0$

"+MMA)=

sin2

r

%

!

The area of a triangle forme# 0$ % ra#ii oflength rof a circle an# the chor# *oining themis given 0$

(here is the angle 0et(een the ra#ii

Abc sin21Cab sin

21 Bca sin

21or or

34 a trig formula for area of a triangle

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