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Homework, Page 484. Solve the triangle. 1. Homework, Page 484. Solve the triangle. 5. Homework, Page 484. Solve the triangle. 9. Homework, Page 484. State whether the given measurements determine zero, one, or two triangles. 13. Homework, Page 484.

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Homework, Page 484

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Homework page 484

Homework, Page 484

Solve the triangle.

1.


Homework page 4841

Homework, Page 484

Solve the triangle.

5.


Homework page 4842

Homework, Page 484

Solve the triangle.

9.


Homework page 4843

Homework, Page 484

State whether the given measurements determine zero, one, or two triangles.

13.


Homework page 4844

Homework, Page 484

State whether the given measurements determine zero, one, or two triangles.

17.


Homework page 4845

Homework, Page 484

Two triangles can be formed using the given measurements. Find both triangles.

21.


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Homework, Page 484

Decide whether the triangle can be solved using the Law of Sines. If so, solve it, if not, explain why not.

25.

Neither triangle can be solved using the Law of Sines, for the one on the left we need to know the length of the side opposite the known angle and for the one on the right, we have the same problem.


Homework page 4847

Homework, Page 484

Respond in one of the following ways:

(a) State: “Cannot be solved with Law of Sines.”

(b) State: “No triangle is formed.”

(c) solve the triangle.

29.

No triangle is formed. The largest side of a triangle is opposite the largest angle and angle A must be the largest angle and side a is no the largest side.


Homework page 4848

Homework, Page 484

Respond in one of the following ways:

(a) State: “Cannot be solved with Law of Sines.”

(b) State: “No triangle is formed.”

(c) solve the triangle.

33.


Homework page 4849

Homework, Page 484

37. Two markers A and B on the same side of a canyon rim are 56 ft apart. A third marker C, located on the opposite rim, is positioned so that

(a) Find the distance between C and A.

(b) Find the distance between the canyon rims.


Homework page 48410

Homework, Page 484

41. A 4-ft airfoil attached to the cab of a truck makes an 18º angle with the roof and angle β is 10º. Find the length of the vertical brace positioned as shown.


Homework page 48411

Homework, Page 484

45. Two lighthouses A and B are known to be exactly 20 mi apart. A ship’s captain at S measures the angle S at 33º. A radio operator measures the angle B at 52º. Find the distance from the ship to each lighthouse.


Homework page 48412

Homework, Page 484

49. The length x in the triangle is

(A) 8.6

(B) 15.0

(C) 18.1

(D) 19.2

(E) 22.6


Homework page 48413

Homework, Page 484

53. (a) Show that there are infinitely many triangles with AAA given if the sum of the three positive angles is 180º.

Consider the triangle formed with its base on a radius that is one-half the diameter of a semi-circle. If the opposite ends of the radius are connected to a point on the semi-circle, a triangle is formed. Since there are an infinite number of possible values of the radius, there must be an infinite number of possible triangles.


Homework page 48414

Homework, Page 484

53. (b) Give three examples of triangles where A = 30º, B = 60º, and C = 90º.

(c) Give three examples where A = B = C = 60º.


Homework page 48415

Homework, Page 484

57. Towers A and B are known to be 4.1 mi apart on level ground. A pilot measures the angles of depression to the towers at 36.5º and 25º, respectively. Find distances AC and BC and the height of the aircraft.


Homework page 484

5.6

The Law of Cosines


Quick review

Quick Review


Quick review solutions

Quick Review Solutions


What you ll learn about

What you’ll learn about

  • Deriving the Law of Cosines

  • Solving Triangles (SAS, SSS)

  • Triangle Area and Heron’s Formula

  • Applications

    … and why

    The Law of Cosines is an important extension of the

    Pythagorean theorem, with many applications.


Deriving the law of cosines

Deriving theLaw of Cosines


Law of cosines

Law of Cosines


Example solving a triangle sas

Example Solving a Triangle (SAS)


Example solving a triangle sss

Example Solving a Triangle (SSS)


Area of a triangle

Area of a Triangle


Heron s formula

Heron’s Formula


Example using heron s formula

Example Using Heron’s Formula

Find the area of a triangle with sides 10, 12, 14.


Example finding the area of a regular circumscribed polygon

Example Finding the Area of a Regular Circumscribed Polygon

Find the area of a regular nonagon (9-sided) circumscribed about a circle of radius 10 in.


Example surveyor s problem

Example Surveyor’s Problem

Tony must find the distance from point A to point B on opposite sides of a lake. He finds point C which is 860 ft from point A and 175 ft from point B. If he measures the angle at point C between points A and B as 78º, what is the distance between points A and B.


Homework

Homework

  • Homework Assignment #1

  • Review Section 5.6

  • Page 494, Exercises: 1 – 53 (EOO)


What you ll learn about1

What you’ll learn about

  • Two-Dimensional Vectors

  • Vector Operations

  • Unit Vectors

  • Direction Angles

  • Applications of Vectors

    … and why

    These topics are important in many real-world applications, such

    as calculating the effect of the wind on an airplane’s path.


Directed line segment

Directed Line Segment


Two dimensional vector

Two-Dimensional Vector


Two dimensional vector1

Two-Dimensional Vector


Initial point terminal point equivalent

Initial Point, Terminal Point, Equivalent


Magnitude

Magnitude


Example finding magnitude of a vector

Example Finding Magnitude of a Vector


Vector addition and scalar multiplication

Vector Addition and Scalar Multiplication


Example performing vector operations

Example Performing Vector Operations


Unit vectors

Unit Vectors


Example finding a unit vector

Example Finding a Unit Vector


Standard unit vectors

Standard Unit Vectors


Resolving the vector

Resolving the Vector


Example finding the components of a vector

Example Finding the Components of a Vector


Example finding the direction angle of a vector

Example Finding the Direction Angle of a Vector


Velocity and speed

Velocity and Speed

The velocity of a moving object is a vector

because velocity has both magnitude and

direction. The magnitude of velocity is speed.


Example writing velocity as a vector

Example Writing Velocity as a Vector


Example calculating the effects of wind velocity

Example Calculating the Effects of Wind Velocity


Example finding the direction and magnitude of the resultant force

Example Finding the Direction and Magnitude of the Resultant Force


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