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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

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

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

17.


Homework, Page 484

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

21.


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 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 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 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 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 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 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 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 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 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.


5.6

The Law of Cosines


Quick Review


Quick Review Solutions


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 theLaw of Cosines


Law of Cosines


Example Solving a Triangle (SAS)


Example Solving a Triangle (SSS)


Area of a Triangle


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

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


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 Assignment #1

  • Review Section 5.6

  • Page 494, Exercises: 1 – 53 (EOO)


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


Two-Dimensional Vector


Two-Dimensional Vector


Initial Point, Terminal Point, Equivalent


Magnitude


Example Finding Magnitude of a Vector


Vector Addition and Scalar Multiplication


Example Performing Vector Operations


Unit Vectors


Example Finding a Unit Vector


Standard Unit Vectors


Resolving the Vector


Example Finding the Components of a Vector


Example Finding the Direction Angle of a Vector


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 Calculating the Effects of Wind Velocity


Example Finding the Direction and Magnitude of the Resultant Force


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