8.1 Non-right Triangles: Law of Sines
3 min read•june 25, 2024
The is a powerful tool for solving . It lets us find missing sides and angles when we don't have enough info for regular trig ratios. This law opens up a whole new world of triangle problem-solving.
We can use the Law of Sines to tackle real-world problems involving distances and angles. It's super handy for things like surveying, navigation, and even figuring out how tall buildings are. Pretty cool stuff!
Law of Sines and Its Applications
Law of Sines for non-right triangles
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- States the ratio of the sine of an angle to the length of the is constant for all three angles and sides in a triangle ABC:
- Applies to all triangles, including oblique triangles
- Solve non-right triangles using Law of Sines when given:
- One angle measure and its opposite side length ( or )
- Measures of two angles ()
- Lengths of two sides and measure of a non- (SSA)
- AAS or ASA solution steps:
- Find the ratio using given angle and side
- Solve for unknown side lengths using this ratio
- If needed, find remaining angle using known angles
- AAA solution steps:
- Set up equation with unknown side lengths using Law of Sines
- Solve for one side length
- Find remaining side lengths using ratio
- SSA may have zero, one, or two possible triangles ():
- Find angle opposite known non-included side using Law of Sines
- Two possible triangles if angle is acute
- One possible triangle if angle is right
- No possible triangles if angle is obtuse or side opposite known angle is shorter than known side adjacent to angle
Area calculation with sine function
- Calculate area of non-right triangle using formula:
- and are lengths of any two sides
- is angle between chosen sides
- Area calculation steps:
- Identify two known side lengths and angle between them
- Substitute values into formula
- Simplify expression to find area
Real-world applications of Law of Sines
- Identify given information in problem and determine if sufficient for Law of Sines
- Sketch triangle diagram, labeling known sides and angles
- Determine applicable case (AAS, ASA, AAA, or SSA)
- Apply appropriate steps to solve for unknown side lengths or angles
- If problem requires area, use formula with known or calculated values
- Interpret results in problem context and provide clear answer with appropriate units (meters, square feet)
Triangle Properties and Trigonometric Relationships
- : Two triangles are congruent if they have the same shape and size
- : Two triangles are similar if they have the same shape but not necessarily the same size
- : Relationships between sides and angles in right triangles, which form the basis for the and other trigonometric functions
- Law of Sines extends these relationships to non-right triangles
Key Terms to Review (17)
AAA: AAA, or the Angle-Angle-Angle criterion, is a method used to determine the congruence of two triangles. It states that if the three angles of one triangle are equal to the three angles of another triangle, then the two triangles are congruent, regardless of the lengths of their sides.
AAS: AAS, or Angle-Angle-Side, is a method used to prove the congruence of two triangles. It states that if two triangles have two corresponding angles and one corresponding side equal, then the triangles are congruent. This theorem is particularly useful in the context of solving non-right triangles using the Law of Sines.
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.
Area = 1/2ab 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 two side lengths (a and b) multiplied by the sine of the included angle (C) between those sides.
ASA: ASA, or Angle-Side-Angle, is a fundamental theorem in trigonometry that establishes a relationship between the angles and sides of a triangle. It is particularly useful in the context of non-right triangles and the application of the Law of Sines.
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 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 Triangles: Non-right triangles are triangles that do not have a 90-degree angle. Unlike right triangles, which have one angle that is a perfect right angle, non-right triangles have all three angles that are less than 90 degrees. These types of triangles are often used in various applications, including surveying, navigation, and architecture.
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.
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 and the application of the Law of Sines, as the presence of an obtuse angle affects the calculations and relationships between the sides and angles of the triangle.
Opposite Side: In the context of non-right triangles and the Law of Sines, the opposite side refers to the side of a triangle that is directly across from a given angle. This side is essential in applying the Law of Sines, which relates the ratios of the sides of a triangle to the sines of the opposite angles.
Sin A / a = sin B / b = sin C / c: This key term represents the Law of Sines, a fundamental relationship in the study of non-right triangles. It states that the ratio of the sine of an angle to the length of its opposite side is constant for all angles and sides of a triangle.
Sine Function: The sine function is a periodic function that describes the y-coordinate of a point moving around the unit circle. It is one of the fundamental trigonometric functions and is widely used in mathematics, physics, and engineering to model various periodic phenomena.
SSA (Side-Side-Angle): SSA, or Side-Side-Angle, is a method used to solve for the unknown sides and angles of a non-right triangle. It involves using the given lengths of two sides and the included angle to determine the remaining unknown values in the triangle.
Triangle Congruence: Triangle congruence refers to the concept of determining when two triangles are exactly the same in size and shape. This is an important topic in geometry as it allows for the comparison and analysis of triangles based on specific criteria.
Triangle Similarity: Triangle similarity is a concept in geometry that describes the relationship between two triangles where the angles of one triangle are congruent to the angles of another triangle, and the sides of one triangle are proportional to the corresponding sides of the other triangle. This concept is particularly relevant in the context of non-right triangles and the application of the Law of Sines.
Trigonometric Ratios: Trigonometric ratios are the fundamental relationships between the sides and angles of a triangle. They are used to describe the geometric properties of triangles, especially in the context of non-right triangles, and are essential for solving problems involving angles, sides, and other trigonometric functions.