Sine, Cosine & Area
Rule and heights &
distances in 2D & 3D
, Area Rule
GIVEN:
an included angle
BASIC FORMULAE
- in any ABC: △ A
Area ABC = 1 bc sin ^A
2
△
c
^
= 1 ac sin B b
2
= 1 ab sin ^C B C
2 a
PROOF
sin ^
B=h and sin ^C = h A
c b
h1
∴ h = c sin ^B and h = b sin ^C c
Area △ABC = 1 ah
b
2
∴ Area △ABC = 1 ac sin ^B = 1 ab sin ^C B
a h2 C
2 2 d
Similarly using AB as the base and h1 as the height
Area △ABC = 1 ac sin ^B = 1 bc sin ^A
2 2
EXAMPLE
A DETERMINE THE AREA OF △ABC
Area = 1 ac sin ^B
2
65 cm
Area = 1 (33) (65) sin 101,7°
101,7°
C 2
B
33 cm
Area = 1050,2 cm2
When working out an angle, we use normal algebra and
sin-1(..)
NB: remember sin has two solutions
Rule and heights &
distances in 2D & 3D
, Area Rule
GIVEN:
an included angle
BASIC FORMULAE
- in any ABC: △ A
Area ABC = 1 bc sin ^A
2
△
c
^
= 1 ac sin B b
2
= 1 ab sin ^C B C
2 a
PROOF
sin ^
B=h and sin ^C = h A
c b
h1
∴ h = c sin ^B and h = b sin ^C c
Area △ABC = 1 ah
b
2
∴ Area △ABC = 1 ac sin ^B = 1 ab sin ^C B
a h2 C
2 2 d
Similarly using AB as the base and h1 as the height
Area △ABC = 1 ac sin ^B = 1 bc sin ^A
2 2
EXAMPLE
A DETERMINE THE AREA OF △ABC
Area = 1 ac sin ^B
2
65 cm
Area = 1 (33) (65) sin 101,7°
101,7°
C 2
B
33 cm
Area = 1050,2 cm2
When working out an angle, we use normal algebra and
sin-1(..)
NB: remember sin has two solutions
