Lesson 8.3: Law of Sines & Law of Cosines Applied Problems

Lesson Description

This lesson extends trigonometry beyond right triangles. Students will solve oblique triangles using the Law of Sines and Law of Cosines, including applications in navigation, surveying, architecture, and physics.

Learning Objectives

• Use the Law of Sines to solve AAS, ASA, and SSA triangle cases.
• Understand and analyze the ambiguous SSA case.
• Use the Law of Cosines for SSS and SAS cases.
• Apply both laws to real-world word problems.

Topics Covered

• Law of Sines formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• Ambiguous Case (SSA)
• Law of Cosines formulas: \[ c^2 = a^2 + b^2 - 2ab\cos C \]
• Choosing the correct law
• Real-world triangle applications

Key Terms

• Oblique Triangle: A triangle with no right angle.
• Law of Sines: Ratio relationship between sides and opposite angles.
• Law of Cosines: Generalization of the Pythagorean Theorem.
• Ambiguous Case (SSA): Situation where two triangles may exist.

Lesson Examples

Example 1: Law of Sines (AAS)

Given \( A = 30^\circ \), \( B = 45^\circ \), and \( a = 10 \). Find side \( b \).

\[ \frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ} \]

\[ b = \frac{10 \sin 45^\circ}{\sin 30^\circ} \]

Example 2: Law of Cosines (SAS)

Given \( a = 7 \), \( b = 9 \), and \( C = 60^\circ \). Find \( c \).

\[ c^2 = 7^2 + 9^2 - 2(7)(9)\cos 60^\circ \]

Example 3: Real-World Application

A surveyor measures two sides of land at 120 m and 150 m with an included angle of 40°. Find the third side using Law of Cosines.

Practice Problems

1. Given \( A = 50^\circ \), \( B = 60^\circ \), and \( a = 12 \). Find side \( b \).
2. Given \( a = 8 \), \( b = 11 \), and \( C = 35^\circ \). Find \( c \).
3. Determine whether one or two triangles exist when \( a = 10 \), \( b = 12 \), and \( A = 30^\circ \).

Tips & Common Mistakes

• Always match each side with its opposite angle.
• Check for the ambiguous SSA case before concluding one solution.
• Use Law of Cosines when you have SAS or SSS information.

Summary

The Law of Sines and Law of Cosines allow us to solve any triangle. These formulas extend trigonometry beyond right triangles and are essential tools in advanced geometry and applied mathematics.