3.1 Law of Sines
■ Solving SAA and ASA Triangles (Case 1)
Data Required for Solving Oblique Triangles
Case 1
_____angles and ________side are known (AAS or ASA).
Case 2
_____sides and _________angle not included between the two sides are known (____________).
This case may lead to more than one triangle.
Case 3
_____ sides are known (____________).
Case 4
_____sides and _____angle ____________between the two sides are known (____________).
If we know three angles of a triangle, we cannot find unique side lengths since AAA assures us only of
similarity, not congruence.
Law of Sines
In any triangle ABC, with sides a, b, and c,
a
b
c


sin A sin B sin C
See Proofs In Mathematics, page 331
That is, according to the law of sines, the lengths of the sides in a triangle are ___________________ to the
sines of the measures of the angles opposite them.
Solving SAA and ASA Triangles (Case 1)
CLASSROOM EXAMPLE 1
(SAA) (Home: copy Text Ex. 1 page 283)
Solve triangle ABC if A = 28.8°, C = 102.6°, and c = 25.3 in.
Answer: a = 12.5 in., B = 48.6°, b = 19.4 in.
CLASSROOM EXAMPLE 2
(ASA) (Home: copy Text Ex. 2 page 283)
Kurt Daniels wishes to measure the distance across the
Gasconade River. See the figure. He determines that
C = 117.2°, A = 28.8°, and b = 75.6 ft. Find the
distance a across the river.
Answer: 65.1ft.
CLASSROOM EXAMPLE 3
(ASA) (Home: copy Text Ex. 7 page 287)
The bearing of a lighthouse from a ship was found to be N 52° W. After the ship sailed 5.8 km due south, the
new bearing was N 23° W. Find the distance, to the nearest tenth kilometer, between the ship and the
lighthouse at each location. Provide sketch.
Answer: first location: 4.7 km; second location: 9.4 km.
Home: Copy Ex. 1, 2, 7 (Sec. 3.1), solve: 5, 9, 45, 49
■ Solving SSA Triangles (Case 2) - The Ambiguous Case of the Law of Sines
Helfpful Notes.
1.
For any angle  of a triangle, 0  sin  1. If sin   1, then   90 and the triangle is a right triangle.
2.
sin  sin 180    (Supplementary angles have the same sine value.)
3.
The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the
middle-valued angle is opposite the intermediate side (assuming the triangle is scalene).
CLASSROOM EXAMPLE 4
Solving the Ambiguous Case (No solutions) (Home: copy Text Ex. 4 page 285)
Solve triangle ABC if a = 17.9 cm, c = 13.2 cm, and C  7530.
CLASSROOM EXAMPLE 5
Solving the Ambiguous Case (Two Triangles) (Home: copy Ex. 5 page 285)
Solve triangle ABC if A = 61.4°, a = 35.5 cm, and b = 39.2 cm.
CLASSROOM EXAMPLE 6
Solving the Ambiguous Case (One Triangle) (Home: copy Ex. 3 page 284)
Solve triangle ABC, given B = 68.7°, b = 25.4 in., and a = 19.6 in.
Home: Copy Ex. 3, 4, 5 (Sec. 3.1), solve: 25, 27, 29, 52, 53
■Area of a Triangle
Area of a Triangle
Area of a Triangle (SAS)
In any triangle ABC, the area
is given by the following formulas.
1
 bc sin A,
2

1
ab sin C ,
2
and
1
 ac sin B
2
Finding the Area of a Triangle (SAS)
Find the area of triangle DEF in the figure.
CLASSROOM EXAMPLE 4
Answer: Area = 43 sq ft
Finding the Area of a Triangle (ASA)
Find the area of triangle ABC if B  5810, a = 32.5 cm, and C  7330.
CLASSROOM EXAMPLE 5
Answer: Area = 576 sq cm