### 7.3

```7
Applications
of
Trigonometry
and Vectors
7.3-1
Applications of Trigonometry
7 and Vectors
7.1 Oblique Triangles and the Law of Sines
7.2 The Ambiguous Case of the Law of
Sines
7.3 The Law of Cosines
7.4 Vectors, Operations, and the Dot
Product
7.5 Applications of Vectors
7.3-2
7.3 The Law of Cosines
Derivation of the Law of Cosines ▪ Solving SAS and SSS
Triangles (Cases 3 and 4) ▪ Heron’s Formula for the Area of a
Triangle
1.1-3
7.3-3
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.
1.1-4
7.3-4
Derivation of the Law of
Cosines
Let ABC be any oblique
triangle located on a
coordinate system as
shown.
The coordinates of A are (x, y). For angle B,
and
Thus, the coordinates of A become (c cos B, c sin B).
7.3-5
Derivation of the Law of
Cosines (continued)
The coordinates of C are (a, 0)
and the length of AC is b.
Using the distance formula, we
have
Square both sides
and expand.
7.3-6
Law of Cosines
In any triangle, with sides a, b, and c,
1.1-7
7.3-7
Note
If C = 90°, then cos C = 0, and the
formula becomes
the
Pythagorean theorem.
1.1-8
7.3-8
Example 1
USING THE LAW OF COSINES IN AN
APPLICATION (SAS)
A surveyor wishes to find the
distance between two
inaccessible points A and B on
opposite sides of a lake. While
standing at point C, she finds that
AC = 259 m, BC = 423 m, and
angle ACB measures 132°40′.
Find the distance AB.
1.1-9
7.3-9
Example 1
USING THE LAW OF COSINES IN AN
APPLICATION (SAS)
Use the law of cosines
because we know the lengths
of two sides of the triangle and
the measure of the included
angle.
The distance between the two points is about 628 m.
1.1-10
7.3-10
Example 2
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SAS)
Solve triangle ABC if A = 42.3°,
b = 12.9 m, and c = 15.4 m.
B < C since it is opposite the shorter of the two sides
b and c. Therefore, B cannot be obtuse.
1.1-11
7.3-11
Example 2
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SAS) (continued)
Use the law of sines to find the
measure of another angle.
≈ 10.47
Now find the measure of the third angle.
1.1-12
7.3-12
Caution
If we used the law of sines to find C
rather than B, we would not have
known whether C is equal to 81.7° or its
supplement, 98.3°.
1.1-13
7.3-13
Example 3
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SSS)
Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and
c = 21.1 ft.
Use the law of cosines to find the measure of the
largest angle, C. If cos C < 0, angle C is obtuse.
Solve for cos C.
1.1-14
7.3-14
Example 3
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SSS) (continued)
Use either the law of sines or the law of cosines to
find the measure of angle B.
Now find the measure of angle A.
1.1-15
7.3-15
Example 4
DESIGNING A ROOF TRUSS (SSS)
Find the measure of angle B in
the figure.
1.1-16
7.3-16
Four possible cases can occur when solving an
oblique triangle.
1.1-17
7.3-17
1.1-18
7.3-18
Heron’s Area Formula (SSS)
If a triangle has sides of lengths a, b, and c,
with semiperimeter
then the area of the triangle is
1.1-19
7.3-19
Example 5
USING HERON’S FORMULA TO FIND
AN AREA (SSS)
The distance “as the crow flies” from Los Angeles to
New York is 2451 miles, from New York to Montreal is
331 miles, and from Montreal to Los Angeles is 2427
miles. What is the area of the triangular region having
these three cities as vertices? (Ignore the curvature of
Earth.)