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10.2 Non-right Triangles: Law of Cosines

3 min read•june 24, 2024

The and are powerful tools for solving non-right triangles. They extend beyond the , allowing us to calculate side lengths and angles in any triangle. These formulas are crucial for real-world applications in surveying, engineering, and astronomy.

Heron's Formula takes it a step further, enabling us to find a triangle's area using only its side lengths. This is especially useful when angles or heights are hard to measure. Together, these concepts form a solid foundation for tackling complex geometric problems in various fields.

Law of Cosines and Heron's Formula

Law of Cosines for non-right triangles

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  • Applies to triangles that are not right triangles, where the Pythagorean theorem is insufficient
  • Relates the lengths of the sides of a triangle to the of one of its angles, providing a more general formula
  • Law of Cosines formula: c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab \cos(C)
    • aa, bb, and cc represent the lengths of the sides of the triangle (, , or )
    • CC represents the angle opposite the side cc (acute, obtuse, or right angle)
  • To solve for a missing side length, substitute the known values into the formula and solve for the unknown side length (algebra)
  • To solve for a missing angle, rearrange the formula to solve for the cosine of the angle
    • Rearranged formula: cos⁡(C)=a2+b2−c22ab\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
    • Use the inverse cosine function ( or cos⁡−1\cos^{-1}) to find the angle CC in degrees or radians
  • Part of a broader field of , which deals with relationships between side lengths and angles of triangles

Real-world applications of Law of Cosines

  • Surveying and navigation: Calculating distances between landmarks or determining positions using
  • Construction and engineering: Designing structures, calculating forces, and determining angles for support beams or roofing
  • Astronomy: Measuring distances between celestial objects and calculating orbits using triangular relationships
  • Solving problems involving non-right triangles in various fields (physics, geography, or computer graphics)
  • Steps to solve real-world problems using Law of Cosines:
    1. Identify the given information and determine which values represent the side lengths and angles of the triangle
    2. Sketch a diagram of the triangle and label the known values to visualize the problem
    3. Determine whether you need to solve for a missing side length or angle based on the given information
    4. Apply the Law of Cosines using the appropriate formula, substituting the known values and solving for the unknown variable
    5. Interpret the result in the context of the problem and provide a meaningful answer or conclusion
  • Often used in conjunction with the for solving oblique triangles

Triangle area using Heron's formula

  • Calculates the area of a triangle using only the lengths of its sides, without requiring the measurement of angles or heights
  • Heron's formula: A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}
    • AA represents the area of the triangle
    • aa, bb, and cc represent the lengths of the sides of the triangle
    • ss represents the , calculated as: s=a+b+c2s = \frac{a+b+c}{2}
  • Steps to use Heron's formula:
    1. Calculate the semi-perimeter ss by adding the lengths of all three sides and dividing by 2
    2. Substitute the values of ss and the side lengths into Heron's formula
    3. Simplify the expression under the square root (multiply factors and combine like terms)
    4. Take the square root of the simplified expression to find the area of the triangle
  • Advantages of Heron's formula: Requires only side lengths, useful when angles or heights are difficult to measure (irregular triangles or real-world situations)
  • : Used in conjunction with the Law of Cosines to solve problems involving forces, velocities, or displacements in physics and engineering
  • Law of Sines: Another important trigonometric formula for solving non-right triangles, often used in combination with the Law of Cosines
  • Oblique triangles: Non-right triangles that can be solved using the Law of Cosines, Law of Sines, or a combination of both

Key Terms to Review (63)

A² = b² + c² - 2bc cos A: The equation a² = b² + c² - 2bc cos A is known as the Law of Cosines, which is a fundamental relationship in trigonometry used to solve for unknown sides or angles in non-right triangles. It connects the lengths of the three sides of a triangle (a, b, and c) to the cosine of one of the angles (A).
Acute Triangle: An acute triangle is a triangle in which all three interior angles are less than 90 degrees. This type of triangle is an essential concept in the study of non-right triangles and the application of the Law of Sines and Law of Cosines.
Angle Addition: Angle addition is a fundamental concept in trigonometry that describes the relationship between the angles of a triangle or other polygons. It is a crucial principle used in the derivation of important trigonometric identities, such as the sum and difference identities covered in the 9.2 section.
Angle Subtraction: Angle subtraction is a trigonometric identity that allows for the simplification of expressions involving the difference between two angles. It is a fundamental concept in the study of 9.2 Sum and Difference Identities, which explores the relationships between trigonometric functions of sums and differences of angles.
Arccosine: Arccosine, also known as the inverse cosine function, is a trigonometric function that allows us to find the angle whose cosine is a given value. It is an essential concept in understanding right triangle trigonometry, inverse trigonometric functions, and the law of cosines for non-right triangles.
C² = a² + b² - 2ab cos C: The formula c² = a² + b² - 2ab cos C is known as the Law of Cosines, which is used to solve for the sides and angles of a non-right triangle when the lengths of two sides and the included angle are given.
Cofunction Identities: Cofunction identities are mathematical relationships that exist between certain trigonometric functions, such as sine, cosine, tangent, and their reciprocal functions (cosecant, secant, and cotangent). These identities allow for the conversion between different trigonometric functions, providing a useful tool for simplifying and solving trigonometric expressions.
Commutative Property: The commutative property is a fundamental mathematical principle that states the order of the operands in an addition or multiplication operation does not affect the result. It allows for the rearrangement of terms without changing the overall value of the expression.
Commutative property of addition: The commutative property of addition states that changing the order of addends does not change the sum. Mathematically, if $a$ and $b$ are real numbers, then $a + b = b + a$.
Complementary Angles: Complementary angles are two angles that add up to 90 degrees. They are angles that, when combined, form a right angle or a perfect square corner.
Cos(A-B): The cosine of the difference between two angles, A and B. This trigonometric identity is useful for simplifying and evaluating expressions involving the cosine function.
Cos^-1: The inverse cosine function, denoted as cos^-1 or arccos, is a trigonometric function that gives the angle whose cosine is a given value. It is used to find the angle when the cosine of that angle is known.
Cosecant: The cosecant, often abbreviated as csc, is one of the fundamental trigonometric functions. It represents the reciprocal of the sine function, meaning it is the ratio of the hypotenuse to the opposite side of a right triangle. The cosecant function is particularly useful in understanding the properties and applications of trigonometry, which are essential in various mathematical and scientific disciplines.
Cosine: Cosine is one of the fundamental trigonometric functions, which describes the ratio between the adjacent side and the hypotenuse of a right triangle. It is a crucial concept in various areas of mathematics, including geometry, algebra, and calculus.
Cotangent: The cotangent is one of the fundamental trigonometric functions, defined as the reciprocal of the tangent function. It represents the ratio of the adjacent side to the opposite side of a right triangle, providing a way to describe the relationship between the sides of a right triangle and the angles formed within it.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
Difference Formulas: Difference formulas are algebraic expressions that describe the relationship between the difference of two trigonometric functions, such as sine, cosine, or tangent. These formulas provide a way to simplify and manipulate expressions involving the difference of trigonometric functions, which is particularly useful in the context of the 9.2 Sum and Difference Identities chapter.
Difference Identity: The difference identity is a fundamental trigonometric identity that relates the difference between two angles to the product of their cosines and sines. It is a key concept in the study of sum and difference identities, which are used to simplify and manipulate trigonometric expressions.
Distributive property: The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. It is expressed as $a(b + c) = ab + ac$.
Distributive Property: The distributive property is a fundamental algebraic rule that allows for the simplification of expressions involving multiplication and addition or subtraction. It states that the product of a number and a sum is equal to the sum of the products of the number with each addend.
Double Angle Formula: The double angle formula is a trigonometric identity that allows for the calculation of the sine, cosine, and tangent of an angle that is twice the value of another angle. These formulas provide a way to simplify and manipulate trigonometric expressions involving double angles.
Equilateral: An equilateral triangle is a polygon with three equal sides and three equal angles. It is a special type of triangle that exhibits symmetry and has all of its sides and angles congruent to one another.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Finite arithmetic sequence: A finite arithmetic sequence is a sequence of numbers with a definite number of terms in which the difference between consecutive terms is constant. This constant difference is known as the common difference.
Heron's Formula: Heron's formula is a mathematical equation used to calculate the area of a triangle given the lengths of its three sides. It provides a way to determine the area of non-right triangles, which is particularly useful in various geometric applications.
Included Angle: The included angle is the angle formed between two sides of a polygon or a triangle. It is the angle that is enclosed or 'included' between the two specified sides.
Isosceles: An isosceles triangle is a triangle with at least two sides of equal length. This type of triangle has two congruent angles and is a special case of a non-right triangle, which is the focus of the Law of Cosines in the context of this chapter.
Law of Cosines: The law of cosines is a fundamental relationship in trigonometry that allows for the calculation of the third side of a triangle given the lengths of the other two sides and the angle between them. It is a generalization of the Pythagorean theorem, which only applies to right triangles.
Law of Sines: The law of sines is a fundamental relationship between the sides and angles of a non-right triangle. It states that the ratio of the length of a side in a triangle to the sine of its opposite angle is constant for all sides and angles of the triangle.
Left-Hand Side: The left-hand side (LHS) refers to the expression or quantity that appears on the left side of an equation or mathematical expression. It represents the known or given information that is being evaluated or manipulated in relation to the right-hand side (RHS) of the equation.
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 the focus of the Law of Sines and Law of Cosines, which provide methods for solving for unknown sides and angles in non-right triangles.
Obtuse Triangle: An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees. This type of triangle is particularly relevant in the context of non-right triangles, where the Law of Sines and the Law of Cosines are applied to solve for unknown sides or angles.
Opposite Side: In the context of non-right triangles and the Law of Cosines, the opposite side refers to the side of the triangle that is opposite the angle being considered. This side is crucial in the application of the Law of Cosines, which relates the lengths of the sides and angles of a triangle.
Periodic Functions: Periodic functions are mathematical functions that repeat their values at regular intervals. This concept is central to the study of trigonometric functions, such as sine and cosine, as well as their applications in various fields, including physics, engineering, and computer science.
Pythagorean Identity: The Pythagorean identity is a fundamental trigonometric identity that relates the trigonometric functions sine, cosine, and tangent. It is a crucial concept in understanding the unit circle and verifying, simplifying, and solving trigonometric expressions and equations.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Quadratic equation: A quadratic equation is a second-degree polynomial equation in a single variable, typically written as $ax^2 + bx + c = 0$, where $a \neq 0$. The solutions to the quadratic equation are known as the roots of the equation.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics.
Radian: A radian is a unit of angle measurement in mathematics, representing the angle subtended by an arc on a circle that is equal in length to the radius of that circle. It is a fundamental unit in trigonometry, providing a way to measure angles that is independent of the size of the circle.
Right-Hand Side: The right-hand side (RHS) of an equation or expression refers to the part of the equation that comes after the equals sign or the operation. It represents the result or the value that is being evaluated or solved for.
SAS Method: The SAS method is a technique used to solve non-right triangles by applying the Law of Cosines. It involves using two sides and the included angle of a triangle to determine the remaining unknown sides and angles.
SAS Triangle: The SAS triangle, or 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 fundamental concept in the context of non-right triangles and the application of the Law of Cosines.
Scalene: A scalene triangle is a triangle in which all three sides have different lengths. This distinguishes it from other types of triangles, such as equilateral and isosceles, where the sides have equal or symmetrical lengths.
Secant: A secant is a line that intersects a circle at two distinct points. It is one of the fundamental trigonometric functions, along with sine, cosine, tangent, and others, that describe the relationships between the sides and angles of a right triangle.
Semi-perimeter: The semi-perimeter of a triangle is defined as half the sum of the lengths of its sides. It is a crucial concept in various calculations related to triangles, particularly when using formulas such as Heron's formula, which allows for the determination of the area of a triangle when the lengths of all three sides are known.
Sequence: A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term.
Simplification: Simplification is the process of reducing the complexity of an expression or equation by applying various mathematical rules and techniques to obtain a simpler, more manageable form. This concept is crucial in the context of rational expressions, rational functions, trigonometric identities, and trigonometric expressions, as it allows for more efficient calculations and better understanding of the underlying mathematical relationships.
Sin(A+B): The sine function of the sum of two angles, A and B, is a fundamental trigonometric identity that describes the relationship between the sides and angles of a right triangle. This identity is crucial in understanding various trigonometric functions and their applications in mathematics, physics, and engineering.
Sine: The sine function, denoted as 'sin', is a trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with cosine and tangent, and is essential in understanding various topics in college algebra.
SSS Method: The SSS method, also known as the Side-Side-Side method, is a technique used to determine the unknown sides and angles of a non-right triangle when the lengths of all three sides are known. This method is particularly relevant in the context of the Law of Cosines, which is a fundamental concept in the study of non-right triangles.
SSS Triangle: The SSS triangle, or Side-Side-Side triangle, is a type of triangle where all three sides are known. This is one of the fundamental ways to determine the unique properties of a triangle, including its angles and area, without any additional information.
Sum Formulas: Sum formulas are mathematical expressions that allow for the simplification of trigonometric expressions involving the sum or difference of two angles. These formulas provide a concise way to evaluate and manipulate trigonometric functions, particularly in the context of 9.2 Sum and Difference Identities.
Sum Identity: The sum identity is a mathematical formula that expresses the sine and cosine of the sum of two angles in terms of the sine and cosine of each individual angle. It helps to simplify complex trigonometric expressions and solve equations involving angles. The sum identities are particularly useful for deriving new values from known angles and understanding the behavior of trigonometric functions in relation to addition.
Supplementary Angles: Supplementary angles are two angles that add up to 180 degrees. They are angles that are adjacent to each other and form a straight line. Understanding supplementary angles is crucial in the context of the 9.2 Sum and Difference Identities, as they are a fundamental concept in trigonometry and are used to derive important identities.
Tangent: A tangent is a straight line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry and geometry, with applications across various mathematical disciplines.
Term: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
Triangulation: Triangulation is a method used in mathematics, particularly in the context of non-right triangles, to determine the unknown sides or angles of a triangle using the known information about the other sides and angles. It is a fundamental concept in the application of the Law of Cosines, which is a key tool for solving non-right triangle problems.
Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationship between the angles and sides of a right triangle. They are widely used in various fields, including engineering, physics, and mathematics, to analyze and solve problems involving periodic phenomena, wave propagation, and the behavior of periodic functions.
Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It provides the tools and techniques to analyze and solve problems involving angles, lengths, and the relationships between them.
Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) of the coordinate plane. It is a fundamental concept in trigonometry, as it provides a visual representation of the relationships between the trigonometric functions and the angles they represent.
Vector Algebra: Vector algebra is a branch of mathematics that deals with the operations and properties of vectors, which are mathematical objects that have both magnitude and direction. It is a fundamental tool in various fields, including physics, engineering, and computer science, where the manipulation and analysis of quantities with both size and direction are crucial.
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