1. Solving oblique (non-right) triangles
Suppose the triangle to be solved does not contain a right angle. If we are given three parts of the triangle, not all angles, then we can solve for the remaining parts.
Usually we label the angles of the triangle as A, B, C and the sides opposite these angles as a, b, and c, respectively.
The Law of Sines given by can be used to solve the following two cases.
Two angles and any side (AAS and ASA)
Two sides and an angle opposite one of them (SSA) C b a A B c

2. An ASA example using the Law of Sines
Two fire stations are located 56.7 miles apart, at points A and B. There is a forest fire at point C. If CAB = 54° and CBA = 58°, which fire station is closer? How much closer?
The fire station at point B is closer to the fire by – = miles. C
68° b a
54°
58° A B
c = 56.7 miles

3. The ambiguous case for solving a triangle (SSA)
Suppose we are given two sides of a triangle and the angle opposite one of them, and we are asked to find the remaining parts of the triangle. Depending on the data given, there may be: two possible triangles, one possible triangle, or no possible triangle.
Example. CAB = 45°, b = 1, and various values of a: C
b=1
a=3/10
No triangle exists.
45° A C C
b=1
One triangle exists.
b=1
a=11/10
a=2/2
45°
45° A B A B C
b=1
Two triangles exist.
a=8/10
45° A

4. The area of an oblique triangle
The area of a triangle is given by the formula
For example, in the triangle below,
By choosing other sides of the triangle as the base, we obtain: The area of any triangle is one-half the product of the lengths of two sides times the sine of the included angle. That is, C b a A B c

5. Solving oblique triangles, continued
Two cases remain for solving an oblique triangle.
Three sides (SSS)
Two sides and their included angle (SAS)
 The Law of Cosines (given below) is used for these cases. C b a A B c
Standard Form
Alternative Form

6. An SAS example using the Law of Cosines
A person leaves her home and walks 5 miles due east and then 3 miles northeast. How far away from home (as the crow flies) is she? As seen from her home, what is the angle  between the easterly direction and the direction to her destination?
By applying the Law of Cosines, we can solve for x:
A application of the alternative form gives:
Destination N x 3
135°
45°
Home 5

7. An SAS example using the Law of Cosines, continued
 Instead of applying the alternative form of the Law of Cosines to find θ, we could have used the Law of Sines, which is an easier computation.
Destination N
7.43 3
135°
45°
Home 5

8. An SSS example using the Law of Cosines
Find the angles of the triangle.
Find the angle opposite the longest side first since it will be the largest angle. This will reduce uncertainty.
Since cos B is negative, B is the triangle's only obtuse angle. B = arccos(– ) = º. Use the Law of Sines to determine A.
C =
An SSS example using the Law of Cosines B 14 8 C A 19

9. Heron's area formula
The Law of Cosines can be used to establish another formula for the area of a triangle.
Heron's Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is
Example. Using Heron's formula, find the area of an equilateral triangle with sides of length In this example, s = 3/2 and s – a = s – b = s – c = 1/2.

10. Solve the triangle--An SSA example.
Given the triangle below (not to scale).
It follows that sin = ______ ,  = ______, and φ= ______. The area of the given triangle is ______ cm2. The length x = _____ cm. φ
10cm
3cm 
110 x

11. How should we solve?
Determine the method needed to solve these triangles. The triangles are not to scale.
(b)
(a)
35º z 16 C y 12 A B θ
55º 18 18
(c)
(d) 10
20º φ θ z z 8 9 φ θ
115º
14.5