SUMMARY CHAPTER 7
Whenever
SAS, ASA, AAS, or SSS is given, the triangle is unique.
Law of Sines
In any
triangle ABC, with sides a, b, and c,
Area of a Triangle
In any
triangle ABC, the area A is
given by the following formulas.
These
formulae use two sides and the angle included between them.
Reminder If
possible, use given values in solving triangles to avoid any rounding errors
from using intermediate values.
Number of Triangles
Satisfying the Ambiguous Case (SSA)
Let sides a and b and angle A be given in triangle ABC.
(The law of sines can be used to calculate the value of sin B.)
1. If applying the
law of sines results in an equation having 
, then
no
triangle satisfies the given conditions.
2. If 
, then one triangle satisfies the given
conditions and 
.
3. If 
, then either one or two triangles satisfy the
given conditions.
a) If 
b) Let 
then a second triangle
exists.
In this case, use 
in the second
triangle.
Law of Cosines
In any
triangle ABC, with sides a, b, and c,
Use the Law
of Cosines to solve SAS and SSS Triangles (Cases 3 & 4)
Summary of
four possible cases when solving an oblique triangle.
|
Oblique Triangle |
Suggest Procedure for Solving |
|
Case
I: One side and two angles are
known. (SAA or ASA) |
Step 1:
Find the remaining angle using sum of angles of triangle. Step
2: Find the remaining sides using the law of
sines. |
|
Case
2: Two sides and one angle (not
included between the two sides) are known. (SSA) |
This is the ambiguous case!! Step 1:
Find an angle using the law of sines Step 2:
Find the remaining angle using sum of the angles of triangle. Step 3:
Find the remaining side using the law of sines. (If a 2nd triangle
exists, repeat) |
|
Case
3: Two sides and the included angle
are known. (SAS) |
Step 1:
Find the third side using the law of cosines. Step 2:
Find the smaller of the two remaining angles using the law of sines. Step 3:
Find the remaining angle using the sum of angles of triangle. |
|
Case4: Three sides are known (SSS) |
Step 1
Find the largest angle using the law of cosines. Step 2
Find either remaining angle using the law of sines. Step
3: Find the
remaining angle using the sum of angles of triangle. |
Heron’s Formula for the Area of a
Triangle
If a
triangle has sides of lengths a, b, and
c, with semi perimeter
then the
area of the triangle is
