scoresvideos
Honors Geometry
Table of Contents
Unit 8 Overview: Right Triangles and Trigonometry
8.1 Pythagorean Theorem and its converse
8.2 Special right triangles
8.3 Trigonometric ratios and solving right triangles
8.4 Law of Sines and Law of Cosines

Oblique triangles are triangles without right angles. The Law of Sines and Law of Cosines are powerful tools for solving these triangles, relating sides and angles in ways that go beyond basic trigonometry.

These laws have real-world applications in surveying, navigation, and architecture. By understanding when to use each law, you can solve complex problems involving distances and angles in non-right triangles.

Solving Oblique Triangles

Law of Sines for oblique triangles

  • Relates the sides and angles of a triangle
    • For a triangle ABC, the Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
      • $a$, $b$, and $c$ represent the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively (scalene triangle)
  • Solve for a missing side length using the Law of Sines:
    • Identify two pairs of known side length and opposite angle measure (given angle and side)
    • Set up the Law of Sines equation using the known pairs
    • Solve for the missing side length (algebraic manipulation)
  • Solve for a missing angle measure using the Law of Sines:
    • Identify two pairs of known side length and opposite angle measure (given two sides and one angle)
    • Set up the Law of Sines equation using the known pairs
    • Solve for the missing angle measure using the inverse sine function to isolate the missing angle (arcsine)

Law of Cosines for side lengths

  • Relates the sides and angles of a triangle
    • For a triangle ABC, the Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cos C$
      • $a$, $b$, and $c$ represent the lengths of the sides, and $C$ is the angle opposite side $c$ (obtuse triangle)
  • Solve for a missing side length using the Law of Cosines:
    • Identify the two known side lengths and the angle between them (given two sides and included angle)
    • Substitute the known values into the Law of Cosines equation
    • Simplify the equation and solve for the missing side length by taking the square root of both sides to isolate the missing side length (quadratic equation)

Applications in real-world problems

  • Identify the given information in the problem
    • Determine the known side lengths and angle measures (surveying, navigation)
  • Sketch a diagram of the oblique triangle
    • Label the known side lengths and angle measures
  • Determine the appropriate law to use based on the given information
    • Use the Law of Sines when given:
      1. Two angles and one side (AAS)
      2. Two sides and a non-included angle (SSA)
    • Use the Law of Cosines when given:
      1. Three sides (SSS)
      2. Two sides and the included angle (SAS)
  • Apply the chosen law to set up an equation
    • Substitute the known values into the equation
  • Solve the equation for the missing side length or angle measure
  • Interpret the solution in the context of the real-world problem (distance between landmarks, height of a building)

Selection of appropriate trigonometric law

  • Law of Sines is appropriate when given:
    • Two angles and one side (AAS)
      • Solve for the missing side length using the known angle and its opposite side (sine function)
    • Two sides and a non-included angle (SSA)
      • Solve for the missing angle using the known side and its opposite angle (inverse sine function)
  • Law of Cosines is appropriate when given:
    • Three sides (SSS)
      • Solve for the missing angle using the three known side lengths (cosine function)
    • Two sides and the included angle (SAS)
      • Solve for the missing side length using the two known sides and the angle between them (cosine function)

Key Terms to Review (20)

Similar triangles: Similar triangles are triangles that have the same shape but may differ in size, meaning their corresponding angles are equal and the lengths of their corresponding sides are in proportion. This property allows for various applications in geometry, including understanding ratios, proportions, and the relationships found within right triangles, as well as extending to applications in trigonometry with the Law of Sines and Law of Cosines.
Scalene Triangle: A scalene triangle is a type of triangle in which all three sides have different lengths, and consequently, all three angles are also different. This unique property of having no equal sides or angles sets scalene triangles apart from other types of triangles, like isosceles and equilateral triangles, and directly influences their angle relationships, compliance with triangle inequalities, and applications of geometric theorems.
Equilateral Triangle: An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are equal, each measuring 60 degrees. This unique property creates a perfect balance in the shape, influencing various geometric concepts such as symmetry, area calculations, and congruence relationships.
Angle-side relationship: The angle-side relationship is a fundamental concept in geometry that describes how the lengths of the sides of a triangle relate to the measures of its angles. In a triangle, larger angles are opposite longer sides, and smaller angles are opposite shorter sides. This relationship is crucial for understanding the properties of triangles and is particularly useful when applying specific mathematical principles.
Finding Missing Angles: Finding missing angles involves determining unknown angle measures within triangles or other geometric figures using mathematical relationships. This concept is crucial for solving problems that involve angles, especially in non-right triangles, where traditional methods may not apply. Understanding how to find these angles allows for better comprehension of geometric properties and relationships, and it plays a key role in applying trigonometric laws effectively.
C² = a² + b² - 2ab*cos(c): This equation represents the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. It is essential for solving triangles that do not meet the criteria for simpler relationships like the Pythagorean theorem, particularly in non-right triangles. By using this formula, you can find unknown side lengths or angles when given certain other sides and angles.
Calculating side lengths: Calculating side lengths refers to the process of determining the lengths of the sides of triangles using specific mathematical relationships. This is particularly important when dealing with non-right triangles, where traditional methods like the Pythagorean theorem do not apply. Techniques such as the Law of Sines and the Law of Cosines are essential for finding unknown side lengths based on known angles and side lengths.
Triangle congruence: Triangle congruence means that two triangles are considered congruent if they have exactly the same size and shape. This concept allows us to use certain criteria to determine whether two triangles are congruent, which is essential in solving problems related to the lengths of sides and measures of angles. Understanding triangle congruence is crucial when applying the Law of Sines and Law of Cosines, as these laws help in finding unknown side lengths or angles in triangles based on congruent relationships.
Acute triangle: An acute triangle is a type of triangle where all three interior angles are less than 90 degrees. This property means that the angles are sharp and pointy, giving the triangle its name. In an acute triangle, the sum of the angles still equals 180 degrees, and this characteristic is crucial for understanding relationships between sides and angles, especially when applying certain mathematical principles.
Law of Sines: The equation $$\frac{a}{\sin(a)} = \frac{b}{\sin(b)} = \frac{c}{\sin(c)}$$ expresses a relationship between the sides and angles of a triangle. This law is used to find unknown angles or sides when dealing with non-right triangles. By understanding this relationship, one can determine various properties of triangles in different scenarios, making it essential for solving problems involving triangle geometry.
Proportionality of Sides: Proportionality of sides refers to the relationship between the lengths of the sides of triangles that are similar or related through certain geometric principles. In the context of triangles, this concept helps establish connections between angles and side lengths, allowing for the application of rules such as the Law of Sines and the Law of Cosines. These laws provide essential tools for solving problems involving triangles, especially when dealing with unknown angles or sides.
Obtuse Triangle: An obtuse triangle is a type of triangle in which one of its angles measures greater than 90 degrees. This defining characteristic sets it apart from acute triangles, which have all angles measuring less than 90 degrees, and right triangles, which contain one angle exactly equal to 90 degrees. The presence of the obtuse angle influences various properties of the triangle, including the relationships between its sides and angles, making it significant in the study of trigonometry and geometry.
Hypotenuse: The hypotenuse is the longest side of a right triangle, opposite the right angle. This key side plays a crucial role in various geometric relationships and calculations, influencing concepts like congruence, trigonometric ratios, and special triangles. Understanding the hypotenuse is essential for applying the Pythagorean theorem and solving real-world problems involving right triangles.
Degrees: Degrees are units of measurement used to quantify angles, indicating how far two lines diverge from a common point. In geometry, degrees help describe the size of angles, which is essential for understanding the properties and relationships of shapes. Degrees also play a crucial role in calculations involving arc length and sector area, as well as in applying the Law of Sines and the Law of Cosines to solve problems involving triangles.
Radians: Radians are a unit of angular measurement used in mathematics, particularly in trigonometry and geometry. They are defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Radians provide a direct relationship between linear and angular measurements, making them particularly useful when calculating arc lengths and areas of sectors, as well as in applying trigonometric laws for non-right triangles.
Law of Cosines: The Law of Cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful in solving triangles that are not right-angled, allowing for the determination of unknown side lengths or angle measures when at least one side length and its adjacent angles are known. This law generalizes the Pythagorean theorem and connects closely with trigonometric ratios, providing essential tools for solving triangles.
Right Triangle: A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This unique feature makes it a fundamental shape in geometry, leading to various important principles and theorems that apply to its properties, such as relationships between the sides and angles, area calculations, and distance computations in coordinate systems.
Opposite side: The opposite side in a triangle is the side that is directly across from a specific angle. This term is crucial when working with triangles, especially when applying trigonometric ratios and laws to find unknown sides or angles. Understanding which side is considered 'opposite' helps in accurately applying functions like sine, cosine, and tangent, as well as using the Law of Sines and Law of Cosines for solving various problems involving triangles.
Isosceles Triangle: An isosceles triangle is a triangle that has at least two sides of equal length, which also means that the angles opposite those sides are equal. This unique property of having two equal sides leads to several important relationships and theorems that are useful in various geometric proofs and calculations.
Adjacent side: An adjacent side is a side of a polygon that is next to another side, specifically in the context of a right triangle, where it refers to the side that forms an angle with the hypotenuse and is not the opposite side. This term is crucial when discussing angles and lengths in right triangles, as it helps define trigonometric ratios like cosine. Understanding adjacent sides is also essential for applying laws related to sine and cosine in more complex triangles.