Glossary

9.3 Solving Oblique Triangles

2 min read•july 25, 2024

Oblique triangles are the wild cards of trigonometry. Unlike their right-angled cousins, they don't play by the same rules. We need special laws to crack their secrets: the Law of Sines and the .

These laws help us solve puzzles about triangles in the real world. From figuring out distances between far-off places to designing sturdy bridges, oblique triangles pop up everywhere. Let's dive into the tricks for taming these tricky shapes.

Understanding Oblique Triangles

Selection of trigonometric laws

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  • Law of Sines used for triangles with two angles and any side (AAS or ASA) or two sides and an angle opposite one (SSA)
  • Law of Cosines applied when given two sides and included angle (SAS) or three sides (SSS)
  • Decision process involves examining given information, matching data to appropriate law, considering ambiguous cases (SSA)

Application of sine and cosine laws

  • Law of Sines formula: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} relates sides to opposite angles
  • Law of Cosines formulas: a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A, b2=a2+c2−2accos⁡Bb^2 = a^2 + c^2 - 2ac \cos B, c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C connect sides and angles
  • Solving process:
    1. Select law matching given information
    2. Calculate unknown sides or angles
    3. Use other law if needed for remaining unknowns
  • Verify solutions by checking angle sum equals 180° and all side lengths positive

Advanced Concepts in Oblique Triangles

Area calculation for oblique triangles

  • formula: Area=12absin⁡CArea = \frac{1}{2}ab \sin C used with two sides and included angle
  • : Area=s(s−a)(s−b)(s−c)Area = \sqrt{s(s-a)(s-b)(s-c)} where s=a+b+c2s = \frac{a+b+c}{2} applied when all three sides known
  • Choose sine formula for two sides and angle, Heron's for all sides
  • Practical applications in land surveying, , and (GPS triangulation)

Problem-solving with oblique triangles

  • Strategy: Identify given info and unknowns, draw diagram, plan solution, execute calculations, interpret results
  • Real-world applications: Navigation (maritime routes), engineering (bridge design), physics (force vectors)
  • Combine concepts: Use area calculations in broader problems (flood risk assessment)
  • Error analysis: Estimate expected ranges, consider physical limitations (maximum building height)
  • Optimization: Maximize or minimize areas or distances (solar panel placement for energy efficiency)

Key Terms to Review (14)

1/2 base times height: The formula '1/2 base times height' is used to calculate the area of a triangle, where 'base' refers to the length of one side of the triangle and 'height' is the perpendicular distance from that side to the opposite vertex. This formula is essential in geometry for understanding how to determine the size of triangles, which are fundamental shapes in various mathematical contexts.
Adjacent side: The adjacent side in a right triangle is the side that forms one of the angles of interest, specifically the angle that is not the right angle. It is important because it helps define the trigonometric ratios, such as cosine and tangent, which relate the lengths of the sides of a triangle to its angles. Understanding this term is crucial for solving triangles and applying trigonometric concepts effectively.
Angle-Angle-Side: Angle-Angle-Side (AAS) is a criterion used to determine the congruence of triangles, stating that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent. This criterion is important because it allows for the establishment of congruence without knowing all three sides, which simplifies the solving process in many scenarios.
Architecture: Architecture is the art and science of designing and constructing buildings and other physical structures, blending functionality with aesthetics. It involves careful planning, creativity, and a deep understanding of spatial relationships, materials, and human needs. Architecture plays a crucial role in shaping our environment and can be closely linked to the application of mathematical concepts, such as those found in trigonometry for accurate measurements and angles.
Complementary Angles: Complementary angles are two angles whose measures add up to exactly 90 degrees. This concept is crucial when dealing with various trigonometric functions and identities, as complementary angles have specific relationships that can simplify calculations and deepen understanding of angle measures in different contexts.
Cosine: Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. This relationship holds true in various contexts, from defining circular functions on the unit circle to solving complex triangles, making it essential in understanding angles and their measures.
Heron's Formula: Heron's Formula is a mathematical equation used to calculate the area of a triangle when the lengths of all three sides are known. This formula is particularly useful for solving triangles that do not have a right angle, allowing for the determination of area without requiring altitude or angles. It connects the side lengths of the triangle to its area, making it an essential tool when applying principles of trigonometry to various problems involving oblique triangles.
Hypotenuse: The hypotenuse is the longest side of a right triangle, opposite the right angle. This side is crucial in trigonometry as it serves as a reference point for defining the other sides in relation to angles, and is essential for applying various trigonometric functions and identities. Understanding the hypotenuse is vital for solving right triangles and analyzing problems involving angles and distances.
Law of Cosines: The Law of Cosines is a mathematical formula used to find a side or angle in any triangle when certain other sides or angles are known. This law extends the Pythagorean theorem, allowing calculations for non-right triangles by relating the lengths of the sides of the triangle to the cosine of one of its angles. It is particularly useful in situations where two sides and the included angle, or all three sides, are known.
Navigation: Navigation refers to the process of determining one's position and planning and following a route, especially in relation to travel across land or water. In mathematics, particularly in trigonometry, navigation utilizes principles like angles, distances, and coordinates to accurately chart courses and solve problems involving triangles, making it essential for mapping and orientation.
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.
Side-side-angle: Side-side-angle (SSA) is a condition in triangle geometry where two sides and a non-included angle are known. This scenario often leads to ambiguous cases where more than one triangle may be formed or no triangle at all. Understanding SSA is crucial for solving oblique triangles as it affects how angles and lengths can be determined based on given information.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. This relationship extends beyond right triangles to various contexts, such as the unit circle and periodic functions, where sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.
Tangent: The tangent function, often abbreviated as 'tan', relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. This function is crucial in various branches of mathematics, especially in solving triangles and modeling periodic phenomena.
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