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The Cosine Law

Source: Triangles - ednal/iStock/Getty Images

Let's Start Thinking

A triangle is the only polygon that cannot be deformed without changing its side lengths.

Sources: Truss - TobinC/iStock/Getty Images; Surveying - SonjaBK/iStock/Getty Images

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Lesson Goals

  • State and prove the cosine law.
  • Use the cosine law to compute an unknown side length in an acute triangle.
  • Use the cosine law to determine the interior angles of an acute triangle.

Try This

Rebecca and Sheldon both observe lightning strike a tree from separate locations. By timing the delay between the lightning strike and the thunder, Rebecca determines that she is \(450\) metres from the tree while Sheldon determines that he is \(510\) metres from the tree. The angle between Rebecca and Sheldon as viewed from the tree is \(35.5^{\circ}\).

  1. Draw and label a triangle with Rebecca, Sheldon, and the tree at its vertices to depict the situation.
  2. What is the distance between Rebecca and Sheldon to the nearest metre?
  3. What are the interior angles in the triangle formed by Rebecca, Sheldon, and the tree to one decimal place of precision?