8-1 chapter 8 applications of trigonometry 8.1 the law of...

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Copyright © 2013 Pearson Education, Inc.

Chapter 8 Applications of Trigonometry

8.1 The Law of Sines

■ Congruency and Oblique Triangles ■ Using the Law of Sines ■ The Ambiguous Case

■ Area of a Triangle

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.

Law of Sines

In any triangle ABC, with sides a, b, and c,

sin sin sin

a b c

A B C

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.

An alternative form of the law of sines is

Using the Law of Sines

Round answers to the nearest tenths, unless directions say otherwise.

EXAMPLE 1 Applying the Law of Sines (SAA)

Solve triangle ABC if A = 32.0°, B = 81.8°, and a = 42.9 cm.

8-2 Chapter 8 Applications of Trigonometry


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 = 112.90°, A = 31.10°, and b = 347.6 ft. Find the

distance a across the river.

Reflect: In Example 2, suppose we are told that A = 31.10°, b = 347.6 ft, and a = 295.4 ft. Can the law of sines

be used to find the distance between A and B? Why or why not?

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. 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 has sides that are all of

different lengths).

EXAMPLE 3 Analyzing Data Involving an Obtuse Angle

Without using the law of sines, explain why A = 104°, a = 26.8 m, and b = 31.3 m cannot be valid for a triangle

ABC.

EXAMPLE 4 Solving the Ambiguous Case (No Such Triangle)

Solve triangle ABC if A = 43.5°, a = 22.8 ft, and b = 42.9 ft.

8-3


EXAMPLE 5 Solving the Ambiguous Case (One Triangle)

Solve triangle ABC, given A = 43.5°, a = 22.8 in., and c = 14.8 in.

EXAMPLE 6 Solving the Ambiguous Case (Two Triangles)

Solve triangle ABC if A = 43.5°, a = 22.8 ft, and b = 24.9 ft.

Area of a Triangle (SAS)

In any triangle ABC, the area is given by the following formulas.

1sin ,

2bc A

1sin ,

2ab C and

1sin

2ac B

That is, the area is half the product of the _____________ of _____________ _____________ and the

_____________ of the angle _____________ between them.



EXAMPLE 8 Finding the Area of a Triangle (SAS)

Find the area of triangle ABC in the figure.

EXAMPLE 9 Finding the Area of a Triangle (ASA)

Find the area of triangle ABC in the figure.

Section 8.2 The Law of Cosines 8-5


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

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.

Law of Cosines

In any triangle ABC, with sides a, b, and c, the following hold.

2 2 2

2 2 2

2 2 2

2 cos ,

2 cos ,

2 cos

a b c bc A

b a c ac B

c a b ab C

That is, according to the law of cosines, the ___________________ of a side of a triangle is equal to the

___________________ of the ___________________ of the other two sides, minus ___________________

___________________ ___________________ of those two sides and the cosine of the angle

___________________ between them.

Reflect: If C = 90°, then what theorem does the law of cosines become?

Using the Law of Cosines

EXAMPLE 2 Applying the Law of Cosines (SAS) Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m.

Reflect: Why couldn’t we use the law of sines to solve Example 2?



EXAMPLE 4 Designing a Roof Truss (SSS) Find angle B to the nearest degree, for the truss shown in the figure.

Oblique Triangle Suggested Procedure for Solving

Case 1: One side and two angles

are known. (SAA or

ASA)

Step 1 Find the remaining angle using

______________________.

Step 2 Find the remaining sides using the

___________ ___________ ___________.

Case 2: Two sides and one angle

(not included between the

two sides) are known.

(SSA)

This is the ambiguous case. There may be no triangle,

one triangle, or two triangles.

Step 1 Find a second angle using the ____________.

Step 2 Find the remaining angle using ____________.

Step 3 Find the remaining sides using the

___________ ___________ ___________.

If two triangles exist, repeat Steps 2 and 3.

Case 3: Two sides and the

included angle are

known. (SAS)

Step 1 Find the third side using the ___________

___________ ___________.

Step 2 Find the smaller of the two remaining angles

using the ___________ ___________

___________.


Case 4: Three sides are known.

(SSS) Step 1 Find the largest angle using the ___________

___________ ___________.

Step 2 Find either remaining angle using the

___________ ___________ ___________.


Section 8.2 The Law of Cosines 8-7


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

1

,2

s a b c

then the area of the triangle is given by the following formula.

s s a s b s c

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 mi, from New York to Montreal is 331

mi, and from Montreal to Los Angeles is 2427 mi. What is the area of the triangular region having these three

cities as vertices? Round to the nearest hundred. (Ignore the curvature of Earth.)



8.3 Vectors, Operations, and the Dot Product

■ Basic Terminology ■ Algebraic Interpretation of Vectors ■ Operations with Vectors

■ Dot Product and the Angle between Vectors

Basic Terminology

Quantities that involve magnitudes, such as 45 lb or 60 mph, can be represented by real numbers called

__________________.

Other quantities, called __________________ __________________, involve both magnitude and direction –

for example, 60 mph east.

A vector quantity can be represented with a directed line segment (a segment that uses an arrowhead to indicate

direction) called a __________________.

The length of the vector represents the __________________ of the vector quantity.

When two letters name a vector, the first indicates the __________________ __________________ and the

second indicates the __________________ __________________of the vector.

Two vectors are equal if and only if they have the same direction and the same magnitude.

The sum of two vectors is also a vector. Vector addition is commutative.

For every vector v there is a vector −v that has the same magnitude as v but opposite direction.

Vector −v is the __________________ of v.

The sum of v and −v has magnitude __________________ and is the __________________

__________________.

Algebraic Interpretation of Vectors

A vector with its initial point at the origin in a rectangular coordinate system is called a __________________

__________________.

A position vector u with its endpoint at the point (a, b) is written , ,a b so

, .a bu

Geometrically a vector is a directed line segment while algebraically it is an ordered pair.

The numbers a and b are the __________________

__________________ and the __________________

__________________, respectively, of vector u.

The figure shows the vector , .a bu The positive angle

between the x-axis and a position vector is the

__________________ __________________ for the vector.

In the figure, __________________is the direction angle for

vector u.

Section 8.8 Parametric Equations, Graphs, and Applications 8-9


Magnitude and Direction Angle of a Vector ,a b

The magnitude (length) of vector ,a bu is given by the following.

2 2u a b

The direction angle satisfies tan ,b

a where 0.a

EXAMPLE 1 Finding Magnitude and Direction Angle

Find the magnitude and direction angle for 3, 2 . u

Horizontal and Vertical Components

The horizontal and vertical components, respectively, of a vector u having magnitude u and direction angle

are the following.

u cosa and u sinb

That is, , cos , sin .a b u u u

EXAMPLE 2 Finding Horizontal and Vertical Components

Vector w in the figure has magnitude 25.0 and direction angle

41.7°. Find the horizontal and vertical components.



EXAMPLE 3 Writing Vectors in the Form ,a b

Write vectors u and v in the figure in the form , .a b

Vector Operations

Let a, b, c, d, and k represent real numbers.

, , ,

, ,

a b c d a c b d

k a b ka kb

If 1 2

u , ,a a then 1 2

u , .a a

, , , , ,a b c d a b c d a c b d

EXAMPLE 5 Performing Vector Operations

Let 2, 1 u and 4, 3 .v See the figure. Find and illustrate

each of the following.

(a) u + v

Addition:

Scalar Multiplication:

Negative:

Subtraction:

Section 8.8 Parametric Equations, Graphs, and Applications 8-11


(b) −2u

(c) 3u − 2v

i, j Form for Vectors

If , ,a bv then , where 1, 0 and 0, 1 .a b v i j i j

Dot Product

The dot product of the two vectors ,a bu and ,c dv is denoted ,u v read

“u dot v,” and given by the following: 𝒖 ∙ 𝒗 = ⟨𝑎, 𝑏⟩ ∙ ⟨𝑐, 𝑑⟩ = 𝑎𝑐 + 𝑏𝑑

The dot product of two vectors is a ____________ ____________, not a vector. The dot product of two

vectors can be positive, 0, or negative.

EXAMPLE 6 Finding Dot Products

Find each dot product.

(a) 2, 3 4, 1 (b) 6, 4 2, 3

Properties of the Dot Product

For all vectors u, v, and w and real numbers k, the following hold.

2

(a) u v v u (b) u v w u v u w

(c) u v w u w v w (d) u v u v u v

(e) 0 u 0 (f) u u u

k k k



Geometric Interpretation of Dot Product

If is the angle between the two nonzero vectors u and v, where 0 180 , then the following holds.

u vcos

u v

For angles between 0° and 180°,cos is positive, 0, or negative when is less than, equal to, or greater

than 90°, respectively. Reflect: What does the sign of the dot product of two vectors tell us about the angle between the two vectors?

EXAMPLE 7 Finding the Angle between Two Vectors

Find the angle between the two vectors.

(a) 3, 4 and 2, 1 u v (b) 2, 6 and 6, 2 u v

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