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6.1. The Law of Sines. Quick Review. Quick Review Solutions. What you’ll learn about. Deriving the Law of Sines Solving Triangles (AAS, ASA) The Ambiguous Case (SSA) Applications … and why The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.

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6.1

The Law of Sines




What you ll learn about l.jpg
What you’ll learn about

  • Deriving the Law of Sines

  • Solving Triangles (AAS, ASA)

  • The Ambiguous Case (SSA)

  • Applications

    … and why

    The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.



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Example Solving a Triangle Given Two Angles and a Side


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Example Solving a Triangle Given Two Angles and a Side


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Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)


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Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)


Example finding the height of a pole l.jpg

A

65º

x

B

15ft

15º

C

Example Finding the Height of a Pole


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A

65º

x

B

15ft

15º

C

Example Finding the Height of a Pole


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6.2

The Law of Cosines




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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.



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Example Solving a Triangle (SAS)


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Example Solving a Triangle (SAS)




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Example Using Heron’s Formula

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


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Example Using Heron’s Formula

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




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Chapter Test

9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?

10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.




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Chapter Test Solutions

9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?

≈0.6 mi

10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.

405π/24≈53.01


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