Download Unit 3 Solving Triangles

HARTFIELD – PRECALCULUS
Unit 3 Solving Triangles
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Solving Right Triangles
Angles of Elevation, Depression, and Inclination
Solving Oblique Triangles; Law of Sines
Law of Cosines
Ambiguous Case of Law of Sines
UNIT 3 NOTES | PAGE 1
This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit.
Know the meanings and uses of these terms:
Angle of Elevation
Angle of Depression
Angle of Inclination
Review the meanings and uses of these terms:
Angle Sum Theorem
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 2
Solving Right Triangles
Given a right triangle, if:
1. the length of two sides are known, or
2. the length of one side and the measure of
one acute angle are known,
then it is possible to find the remaining side
lengths and angle measures.
Each trigonometric ratio is formed by one angle
and two sides. With two pieces of information, it
is possible to create an equation with only one
unknown. The inverse of a trigonometric ratio is
necessary when you are solving for an angle.
When given two sides, find the length of the third
side by the Pythagorean Theorem.
If an acute angle is known, the other acute angle
can be found using the Angle Sum Theorem.
* Angle Sum Theorem: The sum of the angle
measurements in a triangle is 180.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 3
Solve the right triangle. Approximate to two digits
after the decimal.
Ex. 1:
Solve the right triangle. Approximate to two digits
after the decimal.
Ex. 2
7
52
12
8
HARTFIELD – PRECALCULUS
Solve the right triangle. Approximate to two digits
after the decimal.
Ex. 3: The length of the hypotenuse is 15 units
& the larger of the two acute angles
measures 64.
UNIT 3 NOTES | PAGE 4
Sketch the triangle given in example 3.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 5
Angles of Elevation, Depression and Inclination
Definition: An angle of elevation is an angle
above horizontal that an observer
must look to see an object that is
higher than the observer.
Definition: An angle of depression is an angle
below horizontal that an observer
must look to see an object that is
lower than the observer.
Definition: An angle of inclination is an angle
formed by a line above horizontal.
We usually use an angle of inclination in the
context of an object without an observer, such as
the incline of a mountain. However this is not
always the case.
HARTFIELD – PRECALCULUS
Ex. 1: Bill is looking out the window of his third
floor apartment, 20 feet about the ground, as
illustrated at right. (Note, picture is not drawn to scale.)
A: Bill spots a $100 bill on the ground outside his
apartment. If the angle of depression is 18,
how far from Bill’s apartment building is the
money? Approximate to the nearest foot.
B: Bill sees an airplane flying in the distance.
Small planes in this area are usually flying at
10,000 feet above sea level. Bill’s apartment
building is at 1200 feet above sea level. If the
angle of inclination to see the plane is 24, by
line of sight how far away is the airplane?
Approximate to the nearest tenth of mile.
UNIT 3 NOTES | PAGE 6
HARTFIELD – PRECALCULUS
Ex. 2: George is standing near a tree that is
casting a long shadow.
A: The tree stands 40 feet high and the shadow is
100 feet long. At what angle of elevation, with
respect to the ground, is the sun based on this
shadow? Approximate to the nearest tenth of
a degree.
B: George’s eyes are 5 feet above the ground. He
spots a cardinal in the tree and the angle of
elevation for him to see the bird is 32. If
George is 30 feet from the spot beneath the
cardinal, how high up is the bird?
Approximate to the nearest tenth of a foot.
UNIT 3 NOTES | PAGE 7
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 8
Solving Oblique Triangles
Law of Sines
Defintion:
The Law of Sines is a statement of proportionality:
in any triangle each ratio formed by the sine of an
angle to the length of a side opposite the angle is
equal.
An oblique triangle is any triangle
that is not a right triangle; i.e., it does
not have a right angle.
sin A
Given a side length and two additional pieces of
information (side lengths or angle
measurements), it is possible to solve any triangle
for which a solution exists.
a

sin B
b

sin C
c
B
a
When given an oblique triangle with two known
angle measurements, it is possible to use the Law
of Sines to find a unique triangle solution.
c
C
A
Observation:
b
The largest angle of a triangle is
always opposite of the longest
side.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 9
Solve the triangle. Approximate as necessary to
five digits.
Ex. 1:
60
84
25
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 10
Solve the triangle. Approximate as necessary to
five digits.
Ex. 2: b = 1000
A = 22
C = 95
Sketch the triangle from example 2.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 11
Law of Cosines
When given an oblique triangle with all three
sides known or an oblique triangle where the one
known angle is between two known sides, it is
possible to use the Law of Cosines to find a unique
triangle solution.
The Law of Cosines is a set of statements
amending the Pythagorean Theorem such that it
can be applied to any triangle: the square of a side
of a triangle is equal to sum of the squares of the
other sides minus twice the product of the other
sides and the cosine of the first side.
2
2
2
2
2
2
2
2
2
c  a  b  2 ab co s C
a  b  c  2 bc co s A
B
b  c  a  2 ca co s B
a
c
C
A
b
After using the Law of Cosines to find an unknown
side or an unknown angle, it is possible to find the
remaining sides or angles using the Law of Sines.
The Law of Sines should never be used to find an
obtuse angle however; either use the Law of
Cosines or the Angle Sum Theorem.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 12
Solve the triangle. Approximate as necessary to
five digits.
Ex. 1:
55
10
18
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 13
Solve the triangle. Approximate as necessary to
five digits.
Ex. 2: a = 12
c = 30
B = 28
Sketch the triangle from example 2.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 14
Solve the triangle. Approximate as necessary to
five digits.
Ex. 3:
10
20
12
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 15
Solve the triangle. Approximate as necessary to
five digits.
Ex. 4: a = 20
b = 24
c = 32
Sketch the triangle from example 4.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 16
Ambiguous Case of the Law of Sines
If you are given the lengths of two sides and the
measure of angle opposite one of the sides, it is
possible for three scenarios to exist:
1. A unique solution exists for one triangle;
that is, exactly one third side length and
two additional angle measures satisfy the
given information.
2. Two parallel solutions exist, each with a
third side length and two angle measures,
that create triangles satisfying the given
information.
3. No triangle can be formed using the given
information.
The number of triangles satisfying the given
information can be determined based on what
happens when the Law of Sines is applied to find
the measure opposite of one of the side lengths:
1. If an angle measure exists and is less than
the given angle measure, exactly one
triangle satisfies the given information.
2. If an angle measure exists and is of greater
measure than the given angle, two triangles
will satisfy the given information. One
triangle will use the initially found measure
while the second triangle will use the
supplement of the found measure.
3. If no angle measure exists, then no triangle
will satisfy the given information.
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 17
Solve all possible triangles that satisfy the
information given below. Approximate as
necessary to five digits.
Ex.1: a = 16
b = 12
A = 65
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 18
Solve all possible triangles that satisfy the
information given below. Approximate as
necessary to five digits.
Ex. 2: b = 14
c = 12
C = 22
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 19
Solve all possible triangles that satisfy the
information given below. Approximate as
necessary to five digits.
Ex. 3: a = 17
c = 12
C = 79
HARTFIELD – PRECALCULUS
UNIT 3 NOTES | PAGE 20
Solve all possible triangles that satisfy the
information given below. Approximate as
necessary to five digits.
Ex. 4: a = 21
b = 25
A = 29