Glossary

8.2 Non-right Triangles: Law of Cosines

2 min read•june 25, 2024

The 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, 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

Law of cosines for non-right triangles

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  • Solves oblique triangles (non-right triangles) given either:
    • Lengths of two sides and measure of
    • Lengths of all three sides
  • Law of Cosines formulas for triangle ABC with sides aa, bb, cc and angles AA, BB, CC opposite respective sides:
    • a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
    • b2=a2+c2−2accos⁡Bb^2 = a^2 + c^2 - 2ac \cos B
    • c2=a2+b2−2abcos⁡Cc^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 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
    • 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 aa, bb, cc and area AA:
    • A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}
    • ss = of triangle, calculated as s=a+b+c2s = \frac{a+b+c}{2}
  • Finding area using Heron's formula:
    1. Calculate semi-perimeter ss using s=a+b+c2s = \frac{a+b+c}{2}
    2. Substitute values of ss, aa, bb, cc 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 for solving non-right triangles
  • is another important trigonometric relationship for non-right triangles
  • Both laws are essential tools in solving problems involving oblique triangles in various fields

Key Terms to Review (14)

Ambiguous Case: The ambiguous case refers to a situation in trigonometry where the given information is insufficient to uniquely determine the missing side or angle of a non-right triangle. This can occur when applying the Law of Sines or the Law of Cosines.
Cosine Function: The cosine function is a periodic function that describes the x-coordinate of a point on the unit circle as it rotates counterclockwise around the origin. It is one of the fundamental trigonometric functions, along with the sine and tangent functions, and is widely used in various mathematical and scientific applications.
Heron's Formula: Heron's formula is a mathematical equation used to calculate the area of a triangle when the lengths of its three sides are known. It provides a simple and efficient way to determine the area of non-right triangles, which is a crucial concept in the context of the Law of Cosines.
Included Angle: The included angle is the angle between two sides of a non-right triangle. It is a key concept in the study of non-right triangles, particularly in the application of the Law of Sines and the Law of Cosines.
Law of Cosines: The law of cosines is a fundamental trigonometric identity that relates the lengths of the sides of a triangle to the cosine of one of its angles. It provides a way to solve for unknown sides or angles in non-right triangles, which are triangles that do not have a 90-degree angle.
Law of Sines: The law of sines is a fundamental trigonometric relationship that allows for the solution of non-right triangles. It establishes a proportional relationship between the sides and angles of a triangle, enabling the determination of unknown values given sufficient information about the known sides and angles.
Non-right Triangle: A non-right triangle is a triangle in which none of the angles are 90 degrees, or right angles. This type of triangle is the focus of the Law of Cosines, which provides a method for solving for unknown sides or angles in a triangle when the given information does not include a right angle.
Oblique Triangle: An oblique triangle is a triangle in which none of the angles are right angles, meaning all three angles are acute or obtuse. These triangles are commonly encountered in real-world applications and require specialized techniques for their analysis and problem-solving.
SAS Triangle: The SAS triangle, also known as the Side-Angle-Side triangle, is a type of triangle where two sides and the included angle are known. This information is sufficient to uniquely determine the shape and size of the triangle, making it a valuable tool in the context of non-right triangles and the application of the Law of Cosines.
Semi-Perimeter: The semi-perimeter, also known as the semi-circumference, is the half of the perimeter or the total length of the sides of a polygon. It is a fundamental concept in the study of non-right triangles and the application of the Law of Cosines, which is used to solve for unknown sides or angles in these types of triangles.
Solving for a Side: Solving for a side refers to the process of determining the length of an unknown side in a non-right triangle using the Law of Cosines. This technique allows you to find the length of a side given the measurements of the other two sides and the angle between them.
Solving for an Angle: Solving for an angle refers to the process of determining the measure of an unknown angle in a geometric figure, often in the context of non-right triangles. This involves the application of trigonometric relationships and formulas to find the missing angle given other known measurements.
SSS Triangle: The SSS triangle, or Side-Side-Side triangle, is a type of triangle where all three sides are known. This information is crucial in the context of the Law of Cosines, which is used to solve for unknown sides or angles in non-right triangles when the lengths of all three sides are given.
Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental tool in various fields, including engineering, physics, and navigation, and is particularly relevant in the context of non-right triangles and the Law of Cosines.
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