Download Trigonometry 6 - Ambiguous Case_1

Math 20-1
Trigonometry Lesson #6
The Ambiguous Case
Objective: By the end of this lesson, you will be able to:
Recall:
To use the Sine Law, you must know an angle and the _________________ side, as well as one
other __________ or ____________.
If you know the measures of two angles and one side, then the triangle is defined uniquely; that
is, there is only one triangle that satisfies the conditions given.
e.g. 1) Draw ABC with side length b = 6 cm and angles A  30 and C  50 .
However, if you know two sides and the angle opposite one side (SSA), there are three possible
outcomes:
1. There is ______ triangle that satisfies the conditions.
2. There are ______ triangles that satisfy the conditions.
3. There are ______ triangles that satisfy the conditions.
When the given angle is obtuse:
If the given angle (  A ) is obtuse, there are two possible cases:
Case 1: a  b
Case 2: a  b
C
C
b
b
A
A
Math 20-1
Trigonometry Lesson #6
When the given angle is acute:
e.g. 2) Each of the following triangles has A  30 and side b = 4 cm. Draw each of the
following triangles (if possible).
a) a = 2 cm
b) a = 5 cm
C
C
b = 4 cm
b = 4 cm
h
A
h
30
A
c) a = 1.5 cm
30
d) a = 2.5 cm
C
C
b = 4 cm
b = 4 cm
h
A
30
h
A
30
Summary (Given angle is acute):
 There is one triangle that satisfies the given conditions when either:

There are no triangles that satisfy the given conditions when:

There are two triangles that satisfy the given conditions when:
* This is known as the ______________________ ___________.
Math 20-1
Trigonometry Lesson #6
e.g. 3) How many triangles PQR are possible if P  38 , p  4.5" , and q  3.5" ?
e.g. 4) In ABC , A  65 and b  50 cm. Determine the value(s) of a for which there is:
a) one right triangle
b) two possible triangles
If there are two possible triangles, then you can use the Sine Law to solve for both of them. The
angle opposite the longer side will have two possibilities: either acute or obtuse. They will have
the same ___________________ angle.
e.g. 5) In FGH , f  6.5 m, g  5.1 m, and G  41 . Draw a diagram to show the two
possible cases for this triangle, and then determine the measures of the missing sides and
angles for both cases.
Assignment:
p. 108-112 #6, 8-9, 11, 17, 22