5.6: The Sine Law
uppercase for angles
lowercase for sidelenghts
Unit 5: Trigonometric Ratios
MCR3U1: Functions
http://www.dpcdsb.org/AMBRO
Minds On
A
Label the sides of the triangle and state the Sine Law.
The Sine Law can be used when the following are given:
c
b
B
a
OR
C
a
sin A
=
b
sin B
sin A
a
=
sin B
b
c
sin C
=
sin C
=
c
Use the triangle to the left to solve for x. Round your answer to the nearest hundredth.
pair up
sidelenght
with angle
x
112
=
sin 13
sin 61
x sin 61
sin 61
x
=
=
sin 13 112
sin 61
28.81
Action: One Ship, Two Places
From the ground, an airplane can be spotted 4.6 km away at an angle of elevation of 34 o.
The airplane spots a boat 3.9 km away. How far is the boat from the person?
Solving This Problem Manually
A
first find angle "b" and "a" to
IMAGE ONE
be able
to solve for length x
SOLVE FOR X
4.6 km
3.9 km
< a = 180 - (34+41)
3.9
x
=
< a = 105 Angle A
sin 34
sin 105
34
P
3.9
sin 34
x
=
4.6
sin b
3.9 ( sin b ) = 4.6 ( sin 34 )
3.9
3.9
<b
B
= sin ( 0.65956)
3.9 ( sin 105 ) =
IMAGE TWO A
4.6 km
Angle B
105
34
P
x ( sin 34 )
sin 34
3.9 km
sin 34
6.74 m
=
x
41
x
B
<b
= 41
Solving This Problem Using Geometer’s Sketchpad
Question 1: Did your answers above match the distance you calculated manually between
the boat and person.
105
4.6 km
Question 2: Can the boat be any other distance away from the person?
3.9 km
second
boat
location
3.9 km
yes, the boat can be moved at one more spot
34
P
41
new boat
B
Question 3: Rotate point B (side a) clockwise. Can the boat be a different distance away
from the person if it remains 3.9 km from the plane while still on the water horizontal line? If
so at what distance and what angle does B make?
139 and 41 degrees
Question 4: Compare the angle B from question 1 and question 3. What type of angles are
they?
The two angles add up to 180, so they are supplementary angles
Question 5: From the example above calculate the height of the triangle using the
4.6sine
km ratio:
h
h
=
b
sin
A
sin 34 = hyp
2.6 km = h
34
4.6
4.6 ( sin 34 ) = h
What is the height of your triangle on the Geometer’s Sketchpad sketch? Is this value the
Action: The Ambiguous Case Of The Sine Law
a < b sin A: a<h
Change the distance between the boat and the airplane (side a) and make it less than the
altitude of your triangle (side h). Rotate a until a triangle forms with the water. How many
C
h = b sin A
triangles form as you rotate? What type(s) of triangles form?
a
if side a is less than the height,
there are no triangles formed since
side a is too short.
h
b
a
h
A
B
c
a = b sin A
Change the distance between the boat and the airplane (side a) and make it equal to the
altitude of your triangle (side h). Rotate a until a triangle forms with the water. How many
C
triangles form as you rotate? What type(s) of triangles form?
it will form 1 right angle triangle
b sin A < a < b
B
A
Change the distance between the boat and the airplane (side a) and make it greater than
the altitude of your triangle (side h) but less than the distance between the person and plane
(side b). Rotate a until a triangle forms. How many triangles form as you rotate? What
C
C
type(s) of triangles form?
two triangles formed, one obtuse and
one acute
A
a≥b
B
A
Change the distance between the boat and the airplane (side a) and make it greater than or
equal to the distance between the person and the plane (side b). Rotate a until a triangle
forms. How many triangles form as you rotate? What type(s) of triangles form?
C
A
B
B
Summary
Number and Type of Triangles
NO TRIANGLE
Conditions
Rough Sketch
when your "a" is less
than your height.
C
b sin A
a < bsinA
a<h
A
B
ONE RIGHT TRIANGLE
when your "a" is equal to
bsinA
C
a = bsinA
a=h
b sin A
A
TWO TRIANGLES – ONE ACUTE,
ONE OBTUSE
ambiguous case
F
G
B
when your "a" is less than
"b" but greater than bsinA
C
(h<a<b)
b sin A
A
G
F
B
ONE TRIANGLE
when your "a" is greater
than or equal to "b"
C
a<b
b sin A
F
OBTUSE TRIANGLES
Number and Type of Triangles
NO TRIANGLE
Conditions
when "a" is less
than "b"
A
B
Rough Sketch
C
a < b or a = b
A
ONE OBTUSE
when "a" is greater than
"b"
a>b
B
Consolidate: The Ambiguous Case Of The Sine Law
A lighthouse at point L is 10 km from a yacht at point Y and 8 km from a sailboat
at point B. From the yacht, the lighthouse and the sailboat are separated by an
angle of 48o. Determine the distance from the yacht to the sailboat.
First Drawing
Second Drawing
L
10 Km
L
8 Km
10 km
8 km
7.4 km
48
Y
48
x
Y
S
first indication that
it is an ambiguous case:
side "a" (8km) opposite from
the angle is less than side
"b" (10km)
S
use sin law
8
=
10
sin s
sin48
8 (sin s) = 10 (sin 48)
8
8
10 (sin 48)
1. find height
<s
h = bsinA
h = (10)sin48
h = 7.4 km
=
sin (
<s
second indication that this
is an ambiguous case:
=
68 degree
find angle l
the height is less than side a
and side
a is less than side b
<l
<l
h < a <b
7.8
8 10
Third Drawing
this is an ambiguous case,
so now two triangles can be
formed
= 180 - ( 68 + 48 )
= 64 degree
64
10 km
8 km
8
x
=
sin64
48
Pg. 318 #1-3; 4; 6; 8; 10; 12
)
8
sin48
68
x
x (sin48)
sin48
x
=
=
8 (sin64)
sin48
9.7 km