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4.3 –The Ambiguous Case
Mini-Quiz:
1. Determine 𝜃 for the following, if 0 < 𝜃 < 180°
a. 𝑠𝑖𝑛𝜃 = 1
1
b. 𝑐𝑜𝑠𝜃 = 2
Foundations of Math 11
c. 𝑐𝑜𝑠𝜃 = 0
1
d. 𝑠𝑖𝑛𝜃 = 2
The Ambiguous Case (SSA)
 By definition, the word ambiguous means ‘open to two or more interpretations’
 Such is the case for certain solutions when working with the Law of Sines.
o If you are given two angles and one side (ASA or AAS), the Law of Sines
will nicely provide you with ONE solution for a missing side.
o Unfortunately, the Law of Sines has a problem dealing with SSA. If you
are given two sides and one angle (where you must find an angle), the
Law of Sines could possibly provide you with one or more solutions, or
even no solution.
 Before we investigate this situation, there are a few facts we need to
remember.
1.
2.
3.
4.
In a triangle, the sum of the interior angles is 180º.
No triangles can have two obtuse angles.
The sine function has a range of 0 ≤ 𝑠𝑖𝑛𝜃 ≤ 1 .
When sine is positive, the angle lies in either Quadrant I or II.
Examples: Let's look at some cases. In each example, decide whether the given
information points to the existence of one triangle, two triangles or no triangles.
1. If a = 20, c = 16, and A = 30º, how many distinct triangles can be drawn?
C = sin-1 (0.4) = 24º (to the nearest degree) - in Quadrant I.
Sine is also positive in Quadrant II. If we use the reference angle 24º in Quadrant II,
the angle C is 156º.
But, with m<A = 30º and m<C = 156º the sum of the angles would exceed 180º.
Not possible!!!!
Therefore, m<C = 24º, m<A = 30º, and m<B = 126º and only ONE triangle is possible.
Homework: Do: Page 183, #1-3, 6, 8
4.3 –The Ambiguous Case
Foundations of Math 11
2. If a = 7, c = 16, and A = 30º, how many distinct triangles can be drawn?
Since sin C must be < 1, no angle exists for angle C. NO triangle exists for these measurements.
3. If a = 10, b = 16, and A = 30º, how many distinct triangles can be drawn?
B = sin-1(.8) = 53.13010 = 53º. Angles could be 30º, 53º, and 97º : sum 180º
The angle from Quadrant II could create angles 30º, 127º, and 23º : sum 180º
Homework: Do: Page 183, #1-3, 6, 8
4.3 –The Ambiguous Case
4. How many triangles are possible if:
A = 40o, a = 3, b = 4
Foundations of Math 11
D = 120o, d = 2, e = 3
5. Isha is at the starting point of two southeasterly paths at an angle measure of
25° to each other. She followed these directions:
a. Follow the more southerly path for 10.2 km to a lookout.
b. Turn due north and walk another 5.4 km to a cabin.
c. Turn and walk directly back to the starting point along the more
northerly path.
What is the compass direction of the return path, to the nearest degree?
Homework: Do: Page 183, #1-3, 6, 8