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Sine Rule & Cosine Rule
These are two extremely useful trignometric results which are applicable to all triangles, not just right angled ones.
To prove the Sine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C.
Divide each into two right angled triangles.
b
C
b
y
x
b
C
A
A
z
a
a
a
c
c
B
B
x = b sin C
z = a sin C
z = c sin A
y = a sin B
x = b sin A
x = c sin B
∴ b sin C = c sin B
∴ a sin B = b sin A
sin C sin B
=
c
b
sin B sin A
=
b
a
Hence the Sine Rule states
c
sin A sin B sin C
=
=
a
b
c
or
∴ a sin C = c sin A
sin C sin A
=
c
a
a
b
c
=
=
sin A sin B sin C
Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 1
To prove the Cosine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C.
Divide each into two right angled triangles, as per the Sine Rule derivation. However, in this case we will combine some basic trigonometry with
Pythagoras’ theorem to write two expressions for the dividing line x, y or z.
b
b
y
b
C
A
x
a
z
a
a
c
c
c
B
b 2 = ( b cos A ) + y 2
c 2 = ( c cos B ) + x 2
a 2 = ( a cos C ) + z 2
a 2 = ( c − b cos A ) + y 2
b 2 = ( a − c cos B ) + x 2
c 2 = ( b − a cos C ) + z 2
2
2
2
2
∴ b 2 − ( b cos A ) = a 2 − ( c − b cos A )
2
2
2
2
∴ c 2 − ( c cos B ) = b 2 − ( a − c cos B )
2
2
∴ a 2 − ( a cos C ) = c 2 − ( b − a cos C )
2
2
c 2 − c 2 cos 2 B = b 2 − {a 2 − 2ac cos B + c 2 cos 2 B}
a 2 − a 2 cos 2 C = c 2 − {b 2 − 2ab cos C + a 2 cos 2 C}
b 2 − b 2 cos 2 A = a 2 − c 2 + 2bc cos A − b 2 cos 2 A
c 2 − c 2 cos 2 B = b 2 − a 2 + 2ac cos B − c 2 cos 2 B
a 2 − a 2 cos 2 C = c 2 − b 2 + 2ab cos B − a 2 cos 2 C
a 2 = b 2 + c 2 − 2bc cos A
b 2 = c 2 + a 2 − 2ac cos B
c 2 = a 2 + b 2 − 2ab cos C
 b2 + c2 − a 2 
A = cos 

2bc


 c2 + a 2 − b2 
B = cos −1 

2ac


 a2 + b2 − c2 
C = cos −1 

2ab


b 2 − b 2 cos 2 A = a 2 − {c 2 − 2bc cos A + b 2 cos 2 A}
−1
Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 2
Examples:
Use the cosine rule to find angles
given three other sides
For finding angles it is best to use the Cosine Rule, as cosine is single valued in the range 0o... 180o whereas sine has two values.
If the angle is obtuse (i.e. > 90o), then the sine rule can yield an incorrect answer since most calculators will only give the solution to sinθ = k
within the range -90o .... 90o
sin θ =
1
2
cos θ =
1
2
θ = 30o ,150o ,...
θ = 60o ,300o ,...
TWO values in the range 0o... 180o
Which one is correct?
Only one solution in the range 0o... 180o
cosθ
sinθ
is the x coordinate of the unit circle
is the y coordinate of the unit circle
θ
is measured anti-clockwise from the
x axis
Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 3
Surveying example:
Use the measurements below to work out the
height H of Mount Everest above sea level
Surveying example (2):
Note the calculation can also be done more generally, and directly,
using tan
180o − 24o − 135o
= 21o
x
H
135o
45o
24o
10,000m
The plains of Nepal
823m
H
α = 45o
y
h = 823m
Sea level
Sine rule:
x
10, 000
=
o
sin 24
sin 21o
10, 000 sin 24o
∴x =
sin 21o
H − 823 = x sin 45o =
x
2
x
2
10, 000sin 24o
∴ H = 823 +
sin 21o × 2
∴ H = 8848m
∴ H = 823 +
The method of determining altitude of
mountain peaks using elevation
measurements at either end of a flat
baseline was used to great effect
in the Great Trigonometrical Survey of
India which continued for much of the
19th century. George Everest was the
second superintendent of the GTS,
and the world’s highest peak was
(renamed) after him. In Nepal it is
known as Sagarmatha, and in Tibet,
Chomolungma “Mother Goddess of
the Universe”.
β = 24o
x = 10,000m
The plains of Nepal
Sea level
H − h = ( x + y ) tan β
H − h = y tan α
H −h
∴y =
tan α
H −h
∴ H − h = x tan β +
tan β
tan α
tan β
 tan β 
H 1 −
 = h + x tan β − h
tan α
 tan α 
H ( tan α − tan β ) = h tan α + x tan α tan β − h tan β
H = h+
x tan α tan β
tan α − tan β
10, 000 tan 45o tan 24o
∴ H = 823 +
tan 45o − tan 24o
∴ H = 8848m
George Everest
(1790-1866)
Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 4