Transcript Cosine_Rule[1] - gcse-maths

Joan Ridgway
The Cosine Rule:
If the triangle is not right-angled, and there is not
a matching pair, you will need then Cosine Rule.
C
a
b
A
In any triangle ABC
B
c
a  b  c  2bc cos A
2
2
2
Finding the missing side:
A
c 9km
20°
b 12cm
Not to scale
B
xa
C
Is it a right-angled triangle? No
Is there a matching pair?
No
Use the Cosine Rule
Label the sides and angles, calling the
given angle “A” and the missing side “a”.
Finding the missing side:
c 9km
A
20°
b 12cm
Not to scale
B
xa
C
a  b  c  2bc cos A
2
Fill
a 2 in the
12 2bits
 9you
 know.
2 12  9  cos 20
2
2
2
a  12  9  (2 12  9  cos 20)
2
2
2
a  22.026........
a  4.69
x = 4.69cm
Finding the missing side:
C
Not to scale
A man starts at the village of Chartham and walks
5 km due South to Aylesham. Then he walks
5km
another 8 km on a bearing of 130° to Barham.
What is the direct distance between Chartham and
Barham, in a straight line?
A
130°
First, draw a sketch.
8km
Is it a right-angled triangle? No
Is there a matching pair?
No
Use the Cosine Rule
B
C
Finding the missing side:
Not to scale
A man starts at the village of Chartham and walks
5 km due South to Aylesham. Then he walks
5km
another 8 km to on a bearing of 130° to Barham.
b
What is the direct distance between Chartham and
Barham, in a straight line?
A
a  b  c  2bc cos A
2
2
a
130°
2
8km
c
Call
missing
length
to find “a”
the
aLabel
² =the
5²
+other
8² - 2sides
x 5 xyou
8 xwant
cos130°
a² = 25 + 64 - 80cos130°
a² = 140.42
11.85km
a = 11.85
B
Finding the missing angle θ:
C
Not to scale
b
a
6cm
A
9cm
θ°
10cm
c
B
Is it a right-angled triangle? No
Is there a matching pair?
No
Use the Cosine Rule
Label the sides and angles,
calling the missing angle “A”
Finding the missing angle θ:
C
Not to scale
b
a
6cm
A
9cm
θ°
B
10cm
c
a 2  b 2  c 2  2bc cos A
9 2  6 2  10 2  2  6 10  cos 
81  36 100 120  cos
81  136 120  cos
136  81
Shift Cos =
cos  
120
 62.72
cos  0.4583333....
