Trigonometry Revision 3
Core 2 Revision 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
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Chapter 4: The Sine and Cosine Rules
General info. 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
ü Sine and Cosine Rules are used to find the length or angle of a non- right angle triangle. A Ø Side C
Side B C
Side A
B
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2
4
Sine Rule 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find a side length:
a Sin A
b Sin B
c Sin C
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2
4
ü The Sine Rule is NOT in the formula booklet so make sure you learn it, it is not a difficult one.
Sine Rule 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find an angle: Sin A Sin B a b
Sin C c
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§ Remember: ü To find a side length put length on top, ü To find an angle put the angle on top.
Sine Rule- Length example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find a side length: a cm
107º
Not drawn to scale
a Sin 30
6 6 Sin 30 Sin 107 Sin 107 3.14 cm
1
2
Find length a?
30º
6 cm
a Sin A
4
b Sin B
c Sin C
Sine Rule- Angle example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find an angle: B 91º 11.1 cm
A
12.3 cm
Sin A a
Not drawn to scale
For angle A: Sin 91 12.3
b 11.1
For angle C: 91.1+ 64.5= 155.6º 180- 155.6= 24.4º
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Find angle C and A?
C
11.1 Sin91 12.3
64.5º
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4
Sin B Sin C b c
Area of a triangle 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find the area of a triangle the formula is as follows: ½ a b Sin C ü The area of a triangle is not in the ½ b c Sin A formula booklet, so it ½ a c Sin B is important that you
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learn it.
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Area of a triangle- Example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example question of how to calculate the area of a triangle.
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1. A triangle has two lengths of 3 cm and 4 cm and the angle in between is 123º area of a triangle= ½ a b Sin C ½ (3)(4) Sin 123= 5.03cm2
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Cosine Rule 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find the length: a2 = b2 + c2- (2bc Cos A) b2 = a2 + c2- (2bc Cos B) c2 = a2 + b2- (2bc Cos C)
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ü The Cosine Rule is in the formula booklet so you don’t have to learn it.
Cosine Rule 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find an angle: Cos A= b2 + c2 – a2 2bc Cos B = a2 + c2 – b2 2bc
Cos C = a2 + b2 – c2 2bc
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ü The Cosine Rule is given in the formula booklet, but you need to be able to rearrange the formula finding the length to finding the angle.
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4
Cosine Rule- Length example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example to trying to find a length using the cosine rule: A
a2 = b2 + c2- (2bc Cos A) 15.1 cm
C
14.2 cm Not drawn to scale
23º
B
1
2
4
Find the length AC?
15.12 + 14.22 – 2(15.1)(14.2) Cos 23 = 228.01 + 201.64 – 428.84 Cos 23 = 429.65 – 428.84 Cos 23 = 429.65 – 394.75 = Square root of 34.9 = 5.91 cm
Cosine Rule- Angle example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find an angle using the cosine rule: Cos A= b2 + c2 – a2 2bc A 7 cm B 10 cm Not drawn to scale
C
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4
102 + 82 – 72 2(10)(8)
8 cm
1
Find the smallest angle?
115 = 0.72 100
Cos-1 0.72= 44º
100 + 64 - 49 100
Solving triangles 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two sides and an included angle (angle in between the two sides), use Cosine Rule to find the third length. - To find the second angle use Sine Rule - To find the last angle use basic geometry.
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Solving triangles 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two angles and a side, use Sine Rule to find the second side. - To find the last angle, use basic geometry. - To find the last side, use the Sine Rule.
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Solving triangles 3 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given three sides, use the Cosine Rule to find the first angle. - To find the second angle, use the Sine Rule. - To find the last angle, use basic geometry.
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Solving triangles 4 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two sides and a nonincluded angle (angle not between the two given sides), use Sine Rule to find the two angles. - Or to find two possible sides, use the Sine Rule.
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Solving triangles- Example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of solving a triangle: A A
50º 60º B
C
5 cm Not drawn to scale
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Ø Given two angles and a side, so use Sine Rule.
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4
1. To get angle C do: 50 + 60= 110 180- 110= 70º
2. To get another side do: 5 a 5 Sin 60 Sin 50 Sin 60 Sin 50
3. To get the last side do: c 5.65 5.65 Sin 70 Sin 70 Sin 60 Sin 60
= 5.65 cm (Side B)
= 6.13 cm (Side C)
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2
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Chapter 4: The Sine and Cosine Rules
General info. 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
ü Sine and Cosine Rules are used to find the length or angle of a non- right angle triangle. A Ø Side C
Side B C
Side A
B
1
2
4
Sine Rule 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find a side length:
a Sin A
b Sin B
c Sin C
1
2
4
ü The Sine Rule is NOT in the formula booklet so make sure you learn it, it is not a difficult one.
Sine Rule 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find an angle: Sin A Sin B a b
Sin C c
1
2
4
§ Remember: ü To find a side length put length on top, ü To find an angle put the angle on top.
Sine Rule- Length example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find a side length: a cm
107º
Not drawn to scale
a Sin 30
6 6 Sin 30 Sin 107 Sin 107 3.14 cm
1
2
Find length a?
30º
6 cm
a Sin A
4
b Sin B
c Sin C
Sine Rule- Angle example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find an angle: B 91º 11.1 cm
A
12.3 cm
Sin A a
Not drawn to scale
For angle A: Sin 91 12.3
b 11.1
For angle C: 91.1+ 64.5= 155.6º 180- 155.6= 24.4º
1
Find angle C and A?
C
11.1 Sin91 12.3
64.5º
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4
Sin B Sin C b c
Area of a triangle 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find the area of a triangle the formula is as follows: ½ a b Sin C ü The area of a triangle is not in the ½ b c Sin A formula booklet, so it ½ a c Sin B is important that you
1
learn it.
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4
Area of a triangle- Example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example question of how to calculate the area of a triangle.
1
1. A triangle has two lengths of 3 cm and 4 cm and the angle in between is 123º area of a triangle= ½ a b Sin C ½ (3)(4) Sin 123= 5.03cm2
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4
Cosine Rule 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find the length: a2 = b2 + c2- (2bc Cos A) b2 = a2 + c2- (2bc Cos B) c2 = a2 + b2- (2bc Cos C)
1
2
4
ü The Cosine Rule is in the formula booklet so you don’t have to learn it.
Cosine Rule 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ To find an angle: Cos A= b2 + c2 – a2 2bc Cos B = a2 + c2 – b2 2bc
Cos C = a2 + b2 – c2 2bc
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ü The Cosine Rule is given in the formula booklet, but you need to be able to rearrange the formula finding the length to finding the angle.
1
4
Cosine Rule- Length example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example to trying to find a length using the cosine rule: A
a2 = b2 + c2- (2bc Cos A) 15.1 cm
C
14.2 cm Not drawn to scale
23º
B
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2
4
Find the length AC?
15.12 + 14.22 – 2(15.1)(14.2) Cos 23 = 228.01 + 201.64 – 428.84 Cos 23 = 429.65 – 428.84 Cos 23 = 429.65 – 394.75 = Square root of 34.9 = 5.91 cm
Cosine Rule- Angle example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of trying to find an angle using the cosine rule: Cos A= b2 + c2 – a2 2bc A 7 cm B 10 cm Not drawn to scale
C
2
4
102 + 82 – 72 2(10)(8)
8 cm
1
Find the smallest angle?
115 = 0.72 100
Cos-1 0.72= 44º
100 + 64 - 49 100
Solving triangles 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two sides and an included angle (angle in between the two sides), use Cosine Rule to find the third length. - To find the second angle use Sine Rule - To find the last angle use basic geometry.
1
2
4
Solving triangles 2 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two angles and a side, use Sine Rule to find the second side. - To find the last angle, use basic geometry. - To find the last side, use the Sine Rule.
1
2
4
Solving triangles 3 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given three sides, use the Cosine Rule to find the first angle. - To find the second angle, use the Sine Rule. - To find the last angle, use basic geometry.
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2
4
Solving triangles 4 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Ø When you are given two sides and a nonincluded angle (angle not between the two given sides), use Sine Rule to find the two angles. - Or to find two possible sides, use the Sine Rule.
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2
4
Solving triangles- Example 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
§ Here is an example of solving a triangle: A A
50º 60º B
C
5 cm Not drawn to scale
1
Ø Given two angles and a side, so use Sine Rule.
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4
1. To get angle C do: 50 + 60= 110 180- 110= 70º
2. To get another side do: 5 a 5 Sin 60 Sin 50 Sin 60 Sin 50
3. To get the last side do: c 5.65 5.65 Sin 70 Sin 70 Sin 60 Sin 60
= 5.65 cm (Side B)
= 6.13 cm (Side C)
