Honors Precalculus: Oblique Triangles, Law of Sines, and Law of Cosines Study Guide

DEFINITION AND CLASSIFICATION OF OBLIQUE TRIANGLES

• Oblique Triangle Definition: A triangle that is not right-angled is defined as an oblique triangle. This means it does not contain a $90^{\circ}$ angle.

• Classification of Triangles: All triangles can be classified based on the size of their internal angles. These categories include:

  • Acute Triangle: A triangle where all three internal angles are acute (less than $90^{\circ}$).

  • Right Triangle: A triangle containing exactly one right angle ($90^{\circ}$).

  • Obtuse Triangle: A triangle containing exactly one obtuse angle (greater than $90^{\circ}$).

• Nature of Oblique Triangles: Based on the definitions above, oblique triangles are exclusively either acute or obtuse.

• Standard Labeling: In calculations, triangles are typically labeled with vertices $A$, $B$, and $C$. The sides opposite these vertices are denoted by the lowercase letters $a$, $b$, and $c$ respectively.

SOLUTIONS OF OBLIQUE TRIANGLES: CASE CLASSIFICATIONS

To "solve" a triangle means to determine the values of all missing sides and angles. The method used depends on the information provided, categorized into four cases:

• Case 1: [ASA] Angle-Side-Angle scenario: Given two angles and the side included between them. This case is solved using the Law of Sines.

• Case 2: [AAS] Angle-Angle-Side scenario: Given two angles and a side that is not between them. This case is solved using the Law of Sines.

• Case 3: [SAS] Side-Angle-Side scenario: Given two sides and the angle included between them. This case is solved using the Law of Cosines.

• Case 4: [SSS] Side-Side-Side scenario: Given the lengths of all three sides but no angles. This case is solved using the Law of Cosines.

THE LAW OF SINES

• Definition: The Law of Sines is a property used to solve triangles when given AAS or ASA conditions. It establishes a proportional relationship between the side lengths and the sines of their opposite angles.

• The Formula:     $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Application Logic: The Law of Sines is applicable when at least one side and its opposite angle are known, allowing for the calculation of other ratios.

EXERCISE A: LAW OF SINES PROBLEM SETS

• Problem 1: Solving Triangle ABC: Requires finding all missing parts (sides and angles) given specific parameters and sketching the resulting triangle.

• Problem 2: Mountain Peak Elevation (The Matt and Susie Problem):

  • Scenario Description: Matt measures the angle of elevation to a mountain peak as $30^{\circ}$. Susie is on the same straight, level path but is $1200\text{ feet}$ closer to the mountain. She measures the angle of elevation as $75^{\circ}$.

  • Objective: Calculate the height ($h$) of the mountain.

  • Mathematical Context: This involves establishing two triangles or using the external angle of the closer triangle to find the shared side (hypotenuse of the height triangle) using the Law of Sines.

• Problem 3: Aircraft Tracking by Two Observers:

  • Scenario Description: Two observers are positioned exactly $1000\text{ feet}$ apart. An aircraft passes over the line connecting them.

  • Data Points: The first observer records an angle of elevation of $60^{\circ}$. The second observer records an angle of elevation of $15^{\circ}$.

  • Objective: Determine the height of the airplane above the ground.

• Problem 4: Coast Guard SOS Response:

  • Scenario Description: Coast Guard Station Able is exactly $150\text{ miles}$ due South of Station Baker. Both stations receive an SOS call from a ship.

  • Positioning Data:

    • The ship's location relative to Station Able: $\text{N } 30^{\circ} \text{ E}$.

    • The ship's location relative to Station Baker: $\text{S } 45^{\circ} \text{ E}$.

  • Objective (a): Calculate the distance of each station from the ship.

  • Objective (b): Determine the travel time for a rescue helicopter. The helicopter has a speed of $200\text{ miles per hour}$ and is dispatched from the station nearest to the ship.

THE LAW OF COSINES

• Limitations of the Law of Sines: The Law of Sines is insufficient for solving a triangle under two specific conditions:

  • When given the lengths of three sides but no angles (SSS).

  • When given the lengths of two sides and the enclosed angle (SAS).

• Definition: Under SSS or SAS conditions, the Cosine Law (Law of Cosines) must be utilized.

• The Formulas:

  • $a^2 = b^2 + c^2 - 2bc \cos(A)$

  • $b^2 = a^2 + c^2 - 2ac \cos(B)$

  • $c^2 = a^2 + b^2 - 2ab \cos(C)$

EXERCISE B: LAW OF COSINES PROBLEM SETS

• Problem 1: Solving Triangle ABC: Find all missing sides and angles given SAS or SSS information and provide a sketch.

• Problem 2: Multi-Leg Airplane Flight:

  • Flight Path: An airplane flies from Ft. Myers due North to Sarasota for a distance of $75\text{ miles}$. At Sarasota, the pilot turns through an angle of $180$ (as per transcript) and flies to Orlando, a distance of $150\text{ miles}$.

  • Objective (a): Calculate the direct distance from Ft. Myers to Orlando.

  • Objective (b): Determine the angle through which the pilot should turn at Orlando to return directly to Ft. Myers.

• Problem 3: Flight Course Correction (Chicago to Louisville):

  • Original Route: The planned distance from Chicago to Louisville is $330\text{ miles}$.

  • The Error: A pilot inadvertently takes a course that is $50$ (as per transcript) in error from the intended path.

  • The Discovery: The error is discovered after $15\text{ minutes}$ of flying at an average speed of $220\text{ miles per hour}$.

  • Objective (a): Find the angle the pilot must turn to head directly toward Louisville from the current position.

  • Objective (b): Determine the new average speed required to ensure the total trip time remains exactly $90\text{ minutes}$.