10.1 Non-right Triangles: Law of Sines
3 min read•june 24, 2024
The is a powerful tool for solving . It connects the ratios of sines of angles to their opposite lengths, allowing us to find missing angles or sides when given partial information about a triangle.
This law opens up a world of real-world applications, from measuring heights of tall objects to calculating areas of irregular land plots. It's a key concept in that bridges basic triangle geometry with more advanced problem-solving techniques.
Law of Sines and Non-right Triangles
Law of Sines for non-right triangles
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- States the ratio of the of an to the length of the side opposite that angle is constant for all three angles and their corresponding opposite sides in any triangle (including oblique triangles)
- , where , , are angles and , , are the lengths of the sides opposite those angles
- Solve a non-right triangle using the Law of Sines by knowing at least one of these combinations:
- Two angles and any side ()
- Sum of the angles in a triangle is 180° to find the measure of the third angle
- Apply the Law of Sines to find the lengths of the other two sides
- Two sides and an angle opposite one of them ()
- Apply the Law of Sines to find the measure of the angle opposite the other given side
- : there may be two possible triangles, one triangle, or no triangle that satisfies the given conditions
- Two angles and any side ()
Area calculation with sine function
- Calculate the area of a non-right triangle using the sine function and the lengths of two sides
- , where and are the lengths of two sides and is the angle between them
- Find the area by knowing the lengths of two sides and the measure of the included angle ()
- If given other combinations of side lengths and angle measures, use the Law of Sines first to find the required side lengths and angle measures
- Example: given two angles and a side (AAS), find the third angle and apply the Law of Sines to determine the missing side lengths before calculating the area
- If given other combinations of side lengths and angle measures, use the Law of Sines first to find the required side lengths and angle measures
Real-world applications of Law of Sines
- Identify the given information in the problem (side lengths, angle measures)
- Determine the appropriate combination of given information to solve the problem
- AAS: two angles and any side
- SSA: two sides and an angle opposite one of them
- SAS: two sides and the included angle (for area calculation)
- Apply the Law of Sines to find the missing side lengths or angle measures
- Example: finding the height of a tree by measuring the distance from the tree and the angle of elevation to the top of the tree
- If required, calculate the area of the non-right triangle using the sine function
- Example: calculating the area of a triangular plot of land given two side lengths and the included angle
- Interpret the results in the context of the real-world problem
- Consider constraints or limitations based on the problem context (positive side lengths, angle measures between 0° and 180°)
Trigonometry and the Law of Sines
- The Law of Sines is a fundamental concept in trigonometry
- It can be derived using the radius of the of a triangle
- The Law of Sines is particularly useful for solving triangles when given:
- Two angles and a side ()
- Two sides and an angle opposite one of them (SSA)
Key Terms to Review (52)
30-60-90 Triangle: A 30-60-90 triangle is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. This special type of right triangle has unique properties that make it useful in various trigonometric applications.
45-45-90 Triangle: A 45-45-90 triangle is a right triangle in which all three angles are in the ratio of 45 degrees, 45 degrees, and 90 degrees. This unique triangle has special properties that make it particularly useful in various mathematical and geometric applications.
A/sin A = b/sin B = c/sin C: This expression represents the Law of Sines, a fundamental relationship in non-right triangles that connects the side lengths and angle measures. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in the same triangle.
AAS: AAS, or the Angle-Angle-Side theorem, is a fundamental principle in the study of non-right triangles. It provides a method for determining the unknown sides and angles of a triangle given the measure of two angles and one side.
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.
Algebraic Manipulation: Algebraic manipulation refers to the process of performing various operations and transformations on algebraic expressions to simplify, solve, or rearrange them. It involves the strategic use of mathematical rules and properties to manipulate expressions and equations in a logical and systematic manner.
Altitude: Altitude refers to the vertical distance or height of an object above a reference point, typically the ground or sea level. In the context of non-right triangles and the Law of Sines, altitude is a crucial concept that helps determine the relationships between the sides and angles of a triangle.
Ambiguous Case: The ambiguous case refers to a situation in the Law of Sines where the given information is insufficient to uniquely determine the missing parts of a non-right triangle. This occurs when the given information includes one side and two angles, or two sides and one angle, and there are multiple possible solutions for the missing parts.
Angle: An angle is the figure formed by two rays, called the sides of the angle, that share a common endpoint called the vertex. Angles are fundamental geometric concepts that are essential in understanding various mathematical topics, including 7.1 Angles and 10.1 Non-right Triangles: Law of Sines.
Area = 1/2 · ab · sin C: The formula for calculating the area of a non-right triangle using the Law of Sines. It represents the area as half the product of the lengths of two sides (a and b) and the sine of the angle between them (C).
ASA: ASA, which stands for Angle-Side-Angle, is a fundamental congruence criterion used in the context of non-right triangles and the Law of Sines. This criterion establishes that if two triangles have two angles and one corresponding side that are equal, then the triangles are congruent, meaning they are identical in size and shape.
Circumscribed Circle: A circumscribed circle is a circle that passes through all the vertices of a polygon, such as a triangle. It is the smallest circle that can be drawn around a polygon, tangent to each of its sides.
Cofunctions: Cofunctions, in the context of trigonometry, are pairs of trigonometric functions that have a special relationship where one function's value is determined by the other function's value. These cofunctions are closely related and often used together in various trigonometric identities and applications.
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.
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.
Double-Angle Formulas: Double-angle formulas are trigonometric identities that express the sine, cosine, and tangent of twice an angle in terms of the sine and cosine of the original angle. These formulas are crucial for verifying trigonometric identities and simplifying trigonometric expressions involving double-angle terms.
Ellipsis: An ellipsis is a notation consisting of three dots ($\ldots$) used to indicate the continuation of a sequence. It implies that the pattern established by the preceding terms continues in a consistent manner.
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 sequence: A finite sequence is an ordered list of numbers with a specific number of terms. Each term in the sequence has a definite position and can be defined by an explicit formula or pattern.
Fundamental Trigonometric Functions: The fundamental trigonometric functions are the basic trigonometric functions that form the foundation of trigonometry. These functions describe the relationships between the sides and angles of right triangles and are essential for understanding and applying trigonometric identities and simplifying trigonometric expressions.
Half-Angle Formulas: Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half the angle in terms of the sine, cosine, and tangent of the full angle. These formulas are essential for verifying trigonometric identities, simplifying trigonometric expressions, and solving trigonometric equations.
Infinite sequence: An infinite sequence is an ordered list of numbers that continues indefinitely without terminating. Each number in the sequence is called a term, and the position of each term is specified by a positive integer.
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.
N factorial: n factorial, denoted as $n!$, is the product of all positive integers from 1 to $n$. It is commonly used in permutations and combinations to count possible arrangements.
Non-right Triangles: Non-right triangles are triangles that do not have a right angle (an angle measuring 90 degrees). These triangles are important in the context of the Law of Sines, which provides a method for solving for unknown sides and angles in non-right triangles.
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.
Proof: Proof is the process of demonstrating the validity or truth of a mathematical statement or claim through a logical sequence of steps. It involves providing a clear and convincing argument that establishes the veracity of a given proposition or identity.
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.
Quotient Identity: The quotient identity is a fundamental trigonometric identity that relates the sine and cosine functions. It states that the ratio of the sine of an angle to the cosine of the same angle is equal to the tangent of that angle.
Radian Measure: Radian measure is a way of expressing angles in terms of the ratio of the length of the arc subtended by the angle to the radius of the circle. It is a fundamental concept in trigonometry that provides a more natural and versatile way of working with angles compared to the more familiar degree measure.
Rationalization: Rationalization is the process of simplifying or manipulating trigonometric expressions by applying various trigonometric identities and properties. It involves transforming the original expression into an equivalent form that is more easily evaluated or understood.
Reciprocal Identities: Reciprocal identities are a set of fundamental trigonometric identities that express the relationship between the reciprocal trigonometric functions, such as secant, cosecant, and cotangent, and the primary trigonometric functions, such as sine, cosine, and tangent. These identities provide a way to convert between the different trigonometric functions and are essential for verifying and simplifying trigonometric expressions.
SAS: SAS, or Side-Angle-Side, is a fundamental congruence criterion used in the study of non-right triangles. It states that if two triangles have two sides and the included angle equal, then the triangles are congruent, meaning they are identical in size and shape.
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.
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.
Side: In the context of non-right triangles and the Law of Sines, the term 'side' refers to the length of one of the three straight lines that form the perimeter of the triangle. The sides of a triangle are the fundamental elements that define its shape and size, and understanding their relationships is crucial for solving problems involving non-right triangles.
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.
Sin²θ + cos²θ = 1: The trigonometric identity $sin^2\theta + cos^2\theta = 1$ is a fundamental relationship between the sine and cosine functions. It states that the sum of the squares of the sine and cosine of any angle is always equal to 1. This identity is crucial for verifying trigonometric identities and simplifying trigonometric expressions.
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.
SSA: SSA, or Side-Side-Angle, is a fundamental relationship in non-right triangles that describes how the lengths of two sides and the included angle of a triangle can be used to determine the remaining unknown elements of the triangle. This concept is particularly important in the context of the Law of Sines, which provides a method for solving for unknown sides and angles in non-right triangles.
Substitution: Substitution is the process of replacing one or more elements in an expression with other values or variables to simplify, evaluate, or solve the expression. This technique is commonly used in various mathematical contexts, including the composition of functions, verifying trigonometric identities, and solving trigonometric equations.
Sum and Difference Formulas: The sum and difference formulas are a set of trigonometric identities that allow for the simplification and verification of trigonometric expressions involving the addition or subtraction of angles. These formulas are essential in the contexts of verifying trigonometric identities and solving trigonometric equations.
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.
Trigonometric Equations: Trigonometric equations are mathematical equations that involve trigonometric functions, such as sine, cosine, tangent, and their inverse functions. These equations are used to solve for unknown angles or sides of a triangle, or to model periodic phenomena in various fields of study.
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.
Verification: Verification is the process of confirming the accuracy, validity, or truthfulness of a statement, claim, or mathematical identity. It involves demonstrating that a given expression or relationship holds true under the specified conditions or assumptions.