Transcript Finding the missing angle

Joan Ridgway
If the question concerns lengths or angles in a triangle,
you may need the sine rule or the cosine rule.
First, decide if the triangle is right-angled.
Then, decide whether an angle is involved at all.
If it is a right-angled triangle, and there are no angles
involved, you will need Pythagoras’ Theorem
If it is a right-angled triangle, and there are angles
involved, you will need straightforward Trigonometry,
using Sin, Cos and Tan.
If the triangle is not right-angled, you may need the
Sine Rule or the Cosine Rule
The Sine Rule:
Not right-angled!
A
In any triangle ABC
or
C
a
b
c
a
b
c


sin A sin B sin C
sin A sin B sin C


a
b
c
B
You do not have to learn the Sine Rule or the Cosine Rule!
They are always given to you at the front of the Exam Paper.
You just have to know when and how to use them!
The Sine Rule:
C
a
b
A
c
B
You can only use the Sine Rule if you have a “matching pair”.
You have to know one angle, and the side opposite it.
The Sine Rule:
C
a
b
A
c
B
You can only use the Sine Rule if you have a “matching pair”.
You have to know one angle, and the side opposite it.
Then if you have just one other side or angle, you can use
the Sine Rule to find any of the other angles or sides.
Finding the missing side:
Not to scale
10cm
x
40°
65°
Is it a right-angled triangle? No
Is there a matching pair?
Yes
Finding the missing side:
Not to scale
C
b
10cm
A
a
x
40°
c
65°
B
Is it a right-angled triangle? No
Is there a matching pair?
Yes
Use the Sine Rule
Label the sides and angles.
Finding the missing side:
Not to scale
C
b
10cm
A
a
x
40°
c
a
b
c


sin A sin B sin C
65°
B
Because we are trying to find
a missing length of a side,
the little letters are on top
We don’t need the “C” bit of the formula.
Finding the missing side:
Not to scale
C
b
10cm
A
a
x
40°
c
a
b

sin A sin B
Fill in the bits you know.
65°
B
Because we are trying to find
a missing length of a side,
the little letters are on top
Finding the missing side:
Not to scale
C
b
10cm
A
a
x
40°
65°
c
a
b

sin A sin B
B
x
10

sin 40 sin 65
Fill in the bits you know.
Finding the missing side:
Not to scale
C
b
10cm
A
a
x
40°
c
a
b

sin A sin B
65°
B
x
10

sin 40 sin 65
10
x
 sin 40
sin 65
x  7.09 cm
Finding the missing angle:
Not to scale
10cm
7.1cm
θ°
65°
Is it a right-angled triangle? No
Is there a matching pair?
Yes
Finding the missing angle:
Not to scale
C
b
10cm
a
7.1cm
A
θ°
c
65°
B
Is it a right-angled triangle? No
Is there a matching pair?
Yes
Use the Sine Rule
Label the sides and angles.
Finding the missing angle:
Not to scale
C
b
10cm
a
7.1cm
A
θ°
c
sin A sin B sin C


a
b
c
65°
B
Because we are trying to
find a missing angle, the
formula is the other way up.
We don’t need the “C” bit of the formula.
Finding the missing angle:
Not to scale
C
b
10cm
a
7.1cm
A
θ°
c
sin A sin B

a
b
Fill in the bits you know.
65°
B
Because we are trying to
find a missing angle, the
formula is the other way up.
Finding the missing angle:
Not to scale
C
b
10cm
a
7.1cm
A
θ°
65°
c
sin A sin B

a
b
B
sin  sin 65

7 .1
10
Fill in the bits you know.
Finding the missing angle:
Not to scale
C
b
10cm
a
7.1cm
A
θ°
c
sin A sin B

a
b
Shift Sin =
72°
B
sin  sin 65

7 .1
10
sin 65
sin  
 7.1
10
sin   0.6434785.....
  40.05
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 9cm
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 9cm
A
20°
b 12cm
Not to scale
B
xa
C
a  b  c  2bc cos A
a 2  12 2  9 2  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
130°
Barham, in a straight line?
A
2
2
2
a
a  b  c  2bc cos A
Call the missing length you want to find “a”
8km
c
Label the other sides
B
a² = 25 + 64 - 80cos130°
a
²
=
5²
+
8²
2
x
5
x
8
x
cos130°
a² = 140.42
11.85km
a = 11.85
Finding the missing angle A:
C
Not to scale
b
a
6cm
A
9cm
A°
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 A:
C
Not to scale
b
a
6cm
A
9cm
A°
10cm
c
2
B
(b  c  a )
cos A 
(2bc)
2
2
2
(6  10  9 )
cos A 
(2 x6 x10)
2
cos A  0.4583333....
2
Shift Cos =
A  62.72
The Area Rule:
Not right-angled!
A
In any triangle ABC
C
a
b
c
B
1
Area  abSinC
2
Note: the side lengths must surround the angle you use.
Finding the area of this triangle:
C
a 9cm
20°
Not to scale
B
b 12cm
A
Is it a right-angled triangle? No
Do the 2 given sides surround the angle?
Use the Area Rule
Label the sides and angles,
calling the given angle “C”
Yes
Finding the area of this triangle:
C
a 9cm
20°
Not to scale
B
b 12cm
A
Area = ½ abSinC
Area = 0.5 x 9 x 12 x Sin20
Area = 18.469
You Must Not Forget Units!!!!!
18.47cm2