Section 8.1 The Law of Sines
8-1
Chapter 8 Applications of Trigonometry
8.1 The Law of Sines
■ Congruency and Oblique Triangles ■ Derivation of the Law of Sines
■ Using the Law of Sines ■ The Ambiguous Case ■ Area of a Triangle
Key Terms: Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS),
oblique triangle, Side-Angle-Angle (SAA), ambiguous case
Congruency and Oblique Triangles
Congruence Axioms
If two sides and the included angle of one triangle are equal,
Side-Angle-Side
respectively, to two sides and the included angle of a second triangle,
(SAS)
then the triangles are congruent.
Angle-Side-Angle
(ASA)
If two angles and the included side of one triangle are equal,
respectively, to two angles and the included side of a second triangle,
then the triangles are congruent.
Side-Side-Side
(SSS)
If three sides of one triangle are equal, respectively, to three sides of a
second triangle, then the triangles are congruent.
A triangle that is not a right triangle is called an ________________ triangle. The measures
of the three sides and the three angles of a triangle can be found if at least one
________________ and any other ________________ measures are known.
Data Required for Solving Oblique Triangles
Case 1
____________ side and ____________ angles are known (SAA 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 and the angle ____________between the two sides are known
(____________).
Case 4
____________ 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.
Copyright © 2013 Pearson Education, Inc.
8-2
Chapter 8 Applications of Trigonometry
Derivation of the Law of Sines---GSP FILE
Law of Sines
In any triangle ABC, with sides a, b, and c,
a
b
a
c

,

,
sin A sin B sin A sin C
and
b
c

.
sin B sin C
This can be written in compact form as follows.
a
b
c


sin A sin B sin C
An alternative form of the law of sines is


.
Using the Law of Sines
CLASSROOM EXAMPLE 1 Applying the Law of Sines (SAA)
Solve triangle ABC if A = 28.8°, C = 102.6°, and c = 25.3 in.
Copyright © 2013 Pearson Education, Inc.
Section 8.1 The Law of Sines
8-3
CLASSROOM EXAMPLE 2 Applying the Law of Sines (ASA)
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.
The Ambiguous Case
Angle A is
Possible Number of
Triangles
Sketch
Copyright © 2013 Pearson Education, Inc.
Applying Law of
Sines Leads to
8-4
Chapter 8 Applications of Trigonometry
Applying the Law of Sines
1. For any angle  of a triangle, 0  sin  1. If sin   1, then   90 and the triangle is
a right triangle.
2.
3.
sin  sin 180    (Supplementary angles have the same sine value.)
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 has sides that are all of different lengths).
CLASSROOM EXAMPLE 3 Solving the Ambiguous Case (No Such Triangle)
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)
Solve triangle ABC if A = 61.4°, a = 35.5 cm, and b = 39.2 cm.
Copyright © 2013 Pearson Education, Inc.
Section 8.1 The Law of Sines
CLASSROOM EXAMPLE 6 Analyzing Data Involving an Obtuse Angle
Without using the law of sines, explain why no triangle ABC exists satisfying B = 93°,
b = 42 cm, and c = 48 cm.
CLASSROOM EXAMPLE 7 Finding the Area of a Triangle (SAS)
Find the area of triangle DEF in the figure.
Copyright © 2013 Pearson Education, Inc.
8-5
8-6
Chapter 8 Applications of Trigonometry
8.2 The Law of Cosines
■ Derivation of the Law of Cosines ■ Using the Law of Cosines
■ Heron’s Formula for the Area of a Triangle ■ Derivation of Heron’s Formula
Key Terms: semiperimeter
Triangle Side Length Restriction
In any triangle, the sum of the lengths of any two sides must be greater than the length of the
remaining side.
Derivation of the Law of Cosines--GSP
Law of Cosines
In any triangle ABC, with sides a, b, and c, the following hold.
a 2  b 2  c 2  2bc cos A,
b 2  a 2  c 2  2ac cos B ,
c 2  a 2  b 2  2ab cos C
Using the Law of Cosines
CLASSROOM EXAMPLE 1 Applying the Law of Cosines (SAS)
Solve triangle ABC if B = 73.5°, a = 28.2 ft, and c = 46.7 ft.
CLASSROOM EXAMPLE 2 Applying the Law of Cosines (SSS)
Solve triangle ABC if a = 25.4 cm, b = 42.8 cm, and c = 59.3 cm.
Copyright © 2013 Pearson Education, Inc.
Section 8.8 Parametric Equations, Graphs, and Applications
8-7
CLASSROOM EXAMPLE 3 Designing a Roof Truss (SSS)
Find angle C to the nearest degree for the truss shown in the
figure.
Heron’s Formula for the Area of a Triangle
Heron’s Area Formula (SSS)
If a triangle has sides of lengths a, b, and c, with semiperimeter
s
then the area
1
a  b  c,
2
of the triangle is given by the following formula.
 s  s  a  s  b  s  c 
That is, according to Heron’s formula, the area of a triangle is the square root of the
product of four factors: (1) the semiperimeter, (2) the semiperimeter minus the first side,
(3) the semiperimeter minus the second side, and (4) the semiperimeter minus the third
side.
CLASSROOM EXAMPLE 5 Using Heron’s Formula to Find an Area (SSS)
The distance “as the crow flies” from Chicago to St. Louis is 262 mi, from St. Louis to
New Orleans is 599 mi, and from New Orleans to Chicago is 834 mi. What is the area of the
triangular region having these three cities as vertices?
Copyright © 2013 Pearson Education, Inc.