Transcript The Sine Rule. - Stobies Maths Mix
The Sine Rule.
A b o C c o a o B
A
sin
a o
B
sin
b o
C
sin
c o
Finding The Sine Rule.
Consider the triangle below: c o A H B b o a o C Add the altitude line as shown.
H is the height of the triangle. Now write the sine ratio for each right angled triangle:
sin
b o
H A
H = A sin b o
sin
a o
H B
H = B sin a o Look at these two results and try to work out the next line:
c o B A b o a o C H = A sin b o A sin b o = B sin a o H = B sin a o Now divide both sides by sin b o and sin a o .
A
sin
a o
B
sin
b o
By changing the letters around we can prove that:
A
sin
a o
B
sin
b o
C
sin
c o
The Sine Rule.
Calculating Sides Using The
Example 1
Sine Rule.
10m 41 o 34 o L Find the length of L in this triangle.
Match up corresponding sides and angles:
L
sin 41
o
10 sin 34
o
Now cross multiply.
L
sin 34
o
10 sin 41
o L L
10 sin 41 sin 34
o
10
0 .
656
o
11 .
74
m
0 .
559
Solve for L.
Example 2 133 o 10m Find the length of L in this triangle.
37 o L Match up corresponding sides and angles:
L
sin 133
o
10 sin 37
o
Now cross multiply.
L
sin 37
o
10 sin 133
o L
10 sin 133
o
sin 37
o L
10
0 .
731 0 .
602
= 12.14m
Solve for L.
What Goes in the Box ? 1
Find the unknown side in each of the triangles below: B B = 21.8mm
(1) A 72 o 12cm 32 o (2) 47 o A = 6.7cm
93 o 16mm (4) 143 o 17m (3) C 89m 87 o 35 o C = 51.12m
12 o D D = 49.21m
Calculating Angles Using The Sine Rule.
Example 1.
45m Find the angle a o 38m a o 23 o Match up corresponding sides and angles:
45 sin
a o
38 sin 23
o
Now cross multiply:
38 sin
a o
45 sin 23
o
sin
a o
45 sin 23
o
38 = 0.463
Solve for sin a o Use sin -1 0.463 to find a o
a o
sin
1 0 .
463
27 .
6
o
Example 2.
75m Find the size of the angle b o b o 143 o 38m Match up corresponding sides and angles:
38 sin
b o
75 sin 143
o
Cross multiply.
75 sin
b o
38 sin 143
o
Solve for sinb o
b
sin
b o o
38 sin 143
o
sin
1 75 0 .
305 = 0.305
17 .
8
o
Use sin -1 0.305 to find b o
What Goes In The Box ? 2
Calculate the unknown angle in the following: (1) (3) a o 100 o 14.5m
8.9m
a o = 37.2
o (2) 12.9cm
b o 93 o 49mm 14.7cm
14 o c o c =49.9
o 64mm b o = 16 o