9.2 The Law of Cosines
2 min read•july 25, 2024
The Law of Cosines expands on the Pythagorean theorem, allowing us to solve . It's a powerful tool that relates side lengths and , making it possible to tackle more complex geometric problems.
With this law, we can find missing or angles in any triangle. It's super useful in real-world applications like surveying, navigation, and engineering, where right angles aren't always available.
Understanding the Law of Cosines
Law of Cosines fundamentals
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- relates side lengths , , to angle
- Applies to (non-right) expanding Pythagorean theorem's usefulness beyond right triangles
- Solves triangles given two sides and included angle (SAS) or three sides (SSS)
- Calculates both and angles offering more flexibility than Pythagorean theorem
Side length calculations
- Identify known sides and angle, substitute values into formula, solve for unknown side
- Use when given two sides and included angle (SAS) or to verify triangle with all three sides (SSS)
- Algebraically manipulate equation to isolate unknown side, then take square root of both sides
- Ensures accurate side length determination in various triangles (scalene, isosceles)
Angle measurements in triangles
- Rearrange formula to solve for :
- Steps: identify all side lengths, substitute into rearranged formula, solve for , use ()
- Set calculator to , verify result is less than 180°
- Enables angle calculation in triangles where traditional right-angle trigonometry fails
Applications for oblique triangles
- Problem-solving: identify given info, draw labeled diagram, determine solution approach (side or angle)
- Real-world uses: surveying (land boundaries), navigation (GPS positioning), engineering (structural design)
- Consider , appropriate rounding, interpret results contextually
- Combine with Law of Sines, area calculations for comprehensive triangle analysis
- Solve physics problems involving non-perpendicular forces or vectors
Key Terms to Review (18)
Angles: Angles are formed by the intersection of two rays that share a common endpoint, known as the vertex. They are measured in degrees or radians and play a crucial role in the study of triangles, particularly in relation to the lengths of sides and the relationships between them. Understanding angles is essential for applying various trigonometric laws, including those used to solve problems involving triangles and their properties.
Arccos: Arccos, or the inverse cosine function, is used to determine the angle whose cosine is a given number. It is crucial for solving various trigonometric problems, as it allows us to find angles from known cosine values, making it essential for interpreting relationships in right triangles, solving equations, and applying the Law of Cosines.
C² = a² + b² - 2ab * cos(c): This equation, known as the Law of Cosines, relates the lengths of the sides of a triangle to the cosine of one of its angles. It's particularly useful in situations where you need to find an unknown side or angle in a triangle that is not a right triangle. By using this formula, you can solve various problems involving triangles, making it essential for understanding relationships between sides and angles.
Cos(c): The term cos(c) refers to the cosine of an angle 'c', which is a fundamental trigonometric function used to relate the angles and sides of a triangle. It plays a crucial role in various mathematical contexts, particularly in the Law of Cosines, where it helps calculate the lengths of sides or the measures of angles in non-right triangles. Understanding cos(c) allows for deeper insights into the relationships between angles and sides in trigonometric applications.
Degree mode: Degree mode is a setting on a calculator or computing device that allows trigonometric calculations to be performed using degrees as the unit of measurement for angles. This mode is crucial when working with problems involving triangles and circular functions, where angles are often expressed in degrees rather than radians. Understanding how to switch between degree mode and radian mode is essential for accurate calculations in various mathematical contexts.
Inverse cosine: Inverse cosine, often denoted as \(\cos^{-1}(x)\) or \(\text{arccos}(x)\), is a function that returns the angle whose cosine is a given value. This function is essential in determining angles in right triangles and in various applications involving trigonometric ratios, especially when using the Law of Cosines to find an unknown angle from the lengths of the sides of a triangle.
Isosceles Triangle: An isosceles triangle is a type of triangle that has at least two sides of equal length. This property gives rise to specific characteristics, such as two equal angles opposite the equal sides, which makes isosceles triangles unique in their symmetry and properties related to the Law of Cosines.
Law of cosines formula: The law of cosines formula is a fundamental equation in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is essential for solving triangles when given two sides and the included angle, or all three sides, and is particularly useful in non-right triangles. It serves as a generalization of the Pythagorean theorem, allowing for calculations involving any type of triangle.
Non-right triangles: Non-right triangles are triangles that do not contain a 90-degree angle. These triangles can be classified as either acute, where all angles are less than 90 degrees, or obtuse, where one angle is greater than 90 degrees. Understanding non-right triangles is essential for solving various problems in trigonometry, particularly when applying the Law of Cosines to find missing sides or angles.
Oblique triangles: Oblique triangles are triangles that do not contain a right angle, meaning they can either be acute (all angles less than 90 degrees) or obtuse (one angle greater than 90 degrees). Understanding oblique triangles is crucial for solving various real-world problems involving non-right angled shapes and requires different approaches, like the Laws of Sines and Cosines, to find missing sides and angles.
Opposite side: The opposite side in a triangle is the side that is directly across from a specific angle, playing a crucial role in trigonometric calculations. This term is essential for understanding relationships between angles and sides in right triangles and is also significant in solving more complex triangles, as it helps define ratios in trigonometric functions like sine, cosine, and tangent.
Sas condition: The sas condition refers to a specific scenario in triangle geometry where two sides and the included angle of a triangle are known. This condition is essential for applying the Law of Cosines, which allows us to calculate the length of the third side or the angles of the triangle. Understanding the sas condition is crucial for solving problems involving triangles that do not conform to simpler cases like right triangles.
Scalene Triangle: A scalene triangle is a type of triangle where all three sides have different lengths, and consequently, all three angles are also different. This unique feature means that no sides are equal, making it distinct from other types of triangles, such as isosceles and equilateral triangles. The properties of scalene triangles can significantly influence the calculations involved in trigonometry, especially when applying laws like the Law of Cosines to determine unknown lengths or angles.
Sides: In geometry, particularly in the study of triangles, 'sides' refer to the straight line segments that form the boundaries of a triangle. Each triangle has three sides, which can vary in length and are crucial for calculating properties like area, perimeter, and angles. Understanding the relationship between these sides is essential for applying various mathematical laws and solving problems involving triangles.
SSS Condition: The SSS condition refers to the criterion for triangle congruence stating that if all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent. This concept is critical in understanding how the lengths of sides can determine the overall shape and size of triangles, establishing a foundation for more complex geometric principles.
Units consistency: Units consistency refers to the practice of ensuring that all measurements and calculations are expressed in compatible units, which is crucial for accurate mathematical operations and results. When dealing with laws or formulas, like the Law of Cosines, maintaining units consistency allows for the proper evaluation of angles and sides in triangles, facilitating correct applications in real-world scenarios such as physics and engineering.
Unknown angles: Unknown angles refer to angle measures in a triangle or polygon that are not explicitly given and must be determined using mathematical relationships. These angles can be calculated using various methods, including the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles, making it particularly useful when dealing with non-right triangles.
Unknown sides: In geometry and trigonometry, unknown sides refer to the lengths of sides in a triangle that are not given or specified. This term is crucial when using various mathematical tools and methods, such as the Law of Cosines, to determine these lengths based on the information available about the triangle's other sides and angles.