8.2 Non-right Triangles: Law of Cosines

The Law of Cosines is a game-changer for solving non-right triangles. It lets you find missing sides or angles when you don't have a right angle to work with. This opens up a whole new world of triangle-solving possibilities.

Real-world applications abound, from construction to navigation. Plus, Heron's formula lets you find a triangle's area using just the side lengths. These tools are essential for tackling complex geometric problems in various fields.

Law of Cosines and Its Applications

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Law of cosines for non-right triangles

• Solves oblique triangles (non-right triangles) given either:
  • Lengths of two sides and measure of included angle
  • Lengths of all three sides
• Law of Cosines formulas for triangle ABC with sides $a$, $b$, $c$ and angles $A$, $B$, $C$ opposite respective sides:
  • $a^2 = b^2 + c^2 - 2bc \cos A$
  • $b^2 = a^2 + c^2 - 2ac \cos B$
  • $c^2 = a^2 + b^2 - 2ab \cos C$
• Finding unknown side length:
  • Substitute known values into appropriate Law of Cosines formula
  • Solve for unknown side
• Finding unknown angle measure:
  1. Substitute known values into appropriate Law of Cosines formula
  2. Solve for cosine of unknown angle
  3. Take inverse cosine (arccos) of result to find angle measure
• Utilizes the cosine function to relate side lengths and angles in non-right triangles

Real-world applications of law of cosines

• Identify given information (side lengths, angle measures)
• Determine unknown value to find (side length or angle measure)
• Draw diagram representing oblique triangle
  • Label known and unknown values
• Choose appropriate Law of Cosines formula based on unknown value and given information
• Substitute known values into formula
  • Solve for unknown side length or angle measure
• Interpret result in context of real-world problem
  • Construction (building dimensions, roof angles)
  • Navigation (distance between locations, bearing)

Heron's formula for triangle area

• Calculates area of triangle given lengths of all three sides
  • Does not require measurement of angles
• Heron's formula for triangle with sides $a$, $b$, $c$ and area $A$:
  • $A = \sqrt{s(s-a)(s-b)(s-c)}$
  • $s$ = semi-perimeter of triangle, calculated as $s = \frac{a+b+c}{2}$
• Finding area using Heron's formula:
  1. Calculate semi-perimeter $s$ using $s = \frac{a+b+c}{2}$
  2. Substitute values of $s$, $a$, $b$, $c$ into Heron's formula
  3. Simplify and evaluate expression under square root to find area
• Useful when direct measurement of triangle's height is difficult or impractical
  • Irregular shaped plots of land
  • Distances between landmarks

Trigonometry in Non-Right Triangles

• Law of Cosines is a fundamental concept in trigonometry for solving non-right triangles
• Law of Sines is another important trigonometric relationship for non-right triangles
• Both laws are essential tools in solving problems involving oblique triangles in various fields