The Sine And Cosine Rule
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The Sine Rule is used to solve any problems involving triangles when at least either of the following is known: a) two angles and a side b) two sides and an angle opposite a given side
In Triangle ABC, we use the convention that a is the side opposite angle A b is the side opposite angle B A
c
B
b
a
C
The sine rules enables us to calculate sides and angles In the some triangles where there is not a right angle.
Example 2 (Given two sides and an included angle)
<>
Solve triangle ABC in which ∠ A = 55°, b = 2.4cm and c = 2.9cm By cosine rule, a2 = 2.42 + 2.92 - 2 x 2.9 x 2.4 cos 55° = 6.1858 a = 2.49cm
Using this label of a triangle, the sine rule can be stated Either
Or
a b c = = sin A sin B sin C
sin A sin B sin C = = a b c
Use [1] when finding a side Use [2] when finding an angle
[1]
[2]
Example:
A
c
7cm
Given Angle ABC =600 Angle ACB = 500 Find c.
B
C
To find c use the following proportion:
c b = sin C sin B c 7 = sin 50 0 sin 60 0 7 x sin 50 0 c= sin 60 0 c= 6.19 ( 3 S.F)
In ∆BAC AC = 6cm , BC =15 cm and ∠A =120
0
Find ∠B C SOLUTION:
sin B sin A = b a sin B sin 120 0 = 6 15 6 x sin 60 0 sin B = 15 sin B = 0.346
B= 20.30
6 cm 15 cm A
1200
B
DRILL: SOLVE THE FOLLOWING USING THE SINE RULE: Problem 1 (Given two angles and a side) In triangle ABC, ∠ A = 59°, ∠ B = 39° and a = 6.73cm. Find angle C, sides b and c. Problem 2 (Given two sides and an acute angle) In triangle ABC , ∠ A = 55°, b = 16.3cm and a = 14.3cm. Find angle B, angle C and side c.
Problem 3 (Given two sides and an obtuse angle) In triangle ABC ∠ A =100°, b = 5cm and a = 7.7cm Find the unknown angles and side.
Answer Problem 1 ∠ C = 180° - (39° + 59°) = 82°
ANSWER PROBLEM 2
14 .3 16 .3 = 0 sin B sin 55
16.3 c = 0 sin 69 sin 56 0
16.3 sin 550 sin B = 14.3
16 .3 sin 56 0 c= sin 69 0
= 0.9337
= 14.5 cm (3 SF)
∠B = 69.0
0
∠C = 1800 − 69 0 − 550
= 56 0
Answer Problem 3
Sometimes the sine rule is not enough to help u solve for a non-right angled triangle. For example: C
a
14
B
300 18
A
In the triangle shown, we do not have enough information to use the sine rule. That is, the sine rule only provided the Following:
a 14 18 = = 0 sin B sin C sin 30 Where there are too many unknowns.
For this reason we derive another useful result, known as the COSINE RULE. The Cosine Rule maybe used when: a. Two sides and an included angle are given. b. Three sides are given C C
a
A b
B c
A
a
c B
The cosine Rule: To find the length of a side a2 = b2+ c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C
THE COSINE RULE: To find an angle when given all three sides.
b2 +c2 −a2 cos A = 2bc
a2 + c2 − b2 cos B = 2ac
a +b −c cos C = 2ab 2
2
2
Example 1 (Given three sides) In triangle ABC, a = 4cm, b = 5cm and c = 7cm. Find the size of the largest angle. The largest angle is the one facing the longest side, which is angle C.
DRILL:
ANSWER PAGE 203 #’S 1-10
∆
The Sine Rule is used to solve any problems involving triangles when at least either of the following is known: a) two angles and a side b) two sides and an angle opposite a given side
In Triangle ABC, we use the convention that a is the side opposite angle A b is the side opposite angle B A
c
B
b
a
C
The sine rules enables us to calculate sides and angles In the some triangles where there is not a right angle.
Example 2 (Given two sides and an included angle)
<>
Solve triangle ABC in which ∠ A = 55°, b = 2.4cm and c = 2.9cm By cosine rule, a2 = 2.42 + 2.92 - 2 x 2.9 x 2.4 cos 55° = 6.1858 a = 2.49cm
Using this label of a triangle, the sine rule can be stated Either
Or
a b c = = sin A sin B sin C
sin A sin B sin C = = a b c
Use [1] when finding a side Use [2] when finding an angle
[1]
[2]
Example:
A
c
7cm
Given Angle ABC =600 Angle ACB = 500 Find c.
B
C
To find c use the following proportion:
c b = sin C sin B c 7 = sin 50 0 sin 60 0 7 x sin 50 0 c= sin 60 0 c= 6.19 ( 3 S.F)
In ∆BAC AC = 6cm , BC =15 cm and ∠A =120
0
Find ∠B C SOLUTION:
sin B sin A = b a sin B sin 120 0 = 6 15 6 x sin 60 0 sin B = 15 sin B = 0.346
B= 20.30
6 cm 15 cm A
1200
B
DRILL: SOLVE THE FOLLOWING USING THE SINE RULE: Problem 1 (Given two angles and a side) In triangle ABC, ∠ A = 59°, ∠ B = 39° and a = 6.73cm. Find angle C, sides b and c. Problem 2 (Given two sides and an acute angle) In triangle ABC , ∠ A = 55°, b = 16.3cm and a = 14.3cm. Find angle B, angle C and side c.
Problem 3 (Given two sides and an obtuse angle) In triangle ABC ∠ A =100°, b = 5cm and a = 7.7cm Find the unknown angles and side.
Answer Problem 1 ∠ C = 180° - (39° + 59°) = 82°
ANSWER PROBLEM 2
14 .3 16 .3 = 0 sin B sin 55
16.3 c = 0 sin 69 sin 56 0
16.3 sin 550 sin B = 14.3
16 .3 sin 56 0 c= sin 69 0
= 0.9337
= 14.5 cm (3 SF)
∠B = 69.0
0
∠C = 1800 − 69 0 − 550
= 56 0
Answer Problem 3
Sometimes the sine rule is not enough to help u solve for a non-right angled triangle. For example: C
a
14
B
300 18
A
In the triangle shown, we do not have enough information to use the sine rule. That is, the sine rule only provided the Following:
a 14 18 = = 0 sin B sin C sin 30 Where there are too many unknowns.
For this reason we derive another useful result, known as the COSINE RULE. The Cosine Rule maybe used when: a. Two sides and an included angle are given. b. Three sides are given C C
a
A b
B c
A
a
c B
The cosine Rule: To find the length of a side a2 = b2+ c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C
THE COSINE RULE: To find an angle when given all three sides.
b2 +c2 −a2 cos A = 2bc
a2 + c2 − b2 cos B = 2ac
a +b −c cos C = 2ab 2
2
2
Example 1 (Given three sides) In triangle ABC, a = 4cm, b = 5cm and c = 7cm. Find the size of the largest angle. The largest angle is the one facing the longest side, which is angle C.
DRILL:
ANSWER PAGE 203 #’S 1-10
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