📌 Introduction

The Law of Sines and Law of Cosines extend trigonometry beyond right triangles, allowing us to solve any triangle when given sufficient information. These powerful laws are essential for solving oblique (non-right) triangles.

The Formula:

\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Or equivalently:

\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)

When to Use Law of Sines:

📝 Example 1 - Law of Sines (AAS):

In triangle ABC, \( A = 50° \), \( B = 60° \), and \( a = 10 \). Find side \( b \).

Step 1: Set up the ratio:
\( \frac{a}{\sin A} = \frac{b}{\sin B} \)

Step 2: Substitute known values:
\( \frac{10}{\sin 50°} = \frac{b}{\sin 60°} \)

Step 3: Solve for \( b \):
\( b = \frac{10 \times \sin 60°}{\sin 50°} = \frac{10 \times 0.866}{0.766} \approx 11.3 \)

📝 Example 2 - Law of Sines (Finding Angle):

In triangle ABC, \( a = 8 \), \( b = 10 \), and \( A = 40° \). Find angle \( B \).

Step 1: Set up the ratio:
\( \frac{\sin A}{a} = \frac{\sin B}{b} \)

Step 2: Substitute:
\( \frac{\sin 40°}{8} = \frac{\sin B}{10} \)

Step 3: Solve for \( \sin B \):
\( \sin B = \frac{10 \times \sin 40°}{8} = \frac{10 \times 0.643}{8} = 0.804 \)

Step 4: Find \( B \):
\( B = \sin^{-1}(0.804) \approx 53.5° \)

⚠️ The Ambiguous Case (SSA):

When given SSA (two sides and an angle opposite one of them), there may be:

The Formulas:

When to Use Law of Cosines:

Rearranged Forms (to find angles):

📝 Example 1 - Law of Cosines (SAS - Finding Side):

In triangle ABC, \( b = 7 \), \( c = 5 \), and \( A = 60° \). Find side \( a \).

Step 1: Use the formula:
\( a^2 = b^2 + c^2 - 2bc \cos A \)

Step 2: Substitute:
\( a^2 = 7^2 + 5^2 - 2(7)(5) \cos 60° \)
\( a^2 = 49 + 25 - 70(0.5) \)
\( a^2 = 74 - 35 = 39 \)

Step 3: Solve:
\( a = \sqrt{39} \approx 6.2 \)

📝 Example 2 - Law of Cosines (SSS - Finding Angle):

In triangle ABC, \( a = 8 \), \( b = 10 \), and \( c = 12 \). Find angle \( C \).

Step 1: Use the rearranged formula:
\( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \)

Step 2: Substitute:
\( \cos C = \frac{8^2 + 10^2 - 12^2}{2(8)(10)} \)
\( \cos C = \frac{64 + 100 - 144}{160} = \frac{20}{160} = 0.125 \)

Step 3: Find \( C \):
\( C = \cos^{-1}(0.125) \approx 82.8° \)

Strategy for Solving Any Triangle:

📝 Complete Example - Solving a Triangle (SAS):

Given: \( a = 10 \), \( b = 12 \), \( C = 70° \). Find all remaining parts.

Step 1: Find side \( c \) using Law of Cosines:
\( c^2 = a^2 + b^2 - 2ab \cos C \)
\( c^2 = 10^2 + 12^2 - 2(10)(12) \cos 70° \)
\( c^2 = 100 + 144 - 240(0.342) \)
\( c^2 = 244 - 82.1 = 161.9 \)
\( c \approx 12.7 \)

Step 2: Find angle \( A \) using Law of Sines:
\( \frac{\sin A}{a} = \frac{\sin C}{c} \)
\( \sin A = \frac{10 \times \sin 70°}{12.7} = \frac{10 \times 0.940}{12.7} = 0.740 \)
\( A = \sin^{-1}(0.740) \approx 47.7° \)

Step 3: Find angle \( B \):
\( B = 180° - 70° - 47.7° = 62.3° \)

Solution: \( c \approx 12.7 \), \( A \approx 47.7° \), \( B \approx 62.3° \)

Sine Area Formula:

When you know two sides and the included angle:

\( \text{Area} = \frac{1}{2}ab \sin C \)

📝 Example - Area Using Sine:

Find the area of triangle ABC where \( a = 8 \), \( b = 10 \), and \( C = 45° \).

Solution:
\( \text{Area} = \frac{1}{2} \times 8 \times 10 \times \sin 45° \)
\( = \frac{1}{2} \times 8 \times 10 \times 0.707 \)
\( = 40 \times 0.707 \)
\( \approx 28.3 \) square units

When Given Different Information:

If you don't have the included angle, you can:

📝 Example - Area Using Law of Sines:

Find the area of triangle ABC where \( a = 12 \), \( A = 40° \), and \( B = 70° \).

Step 1: Find angle \( C \):
\( C = 180° - 40° - 70° = 70° \)

Step 2: Find side \( b \) using Law of Sines:
\( \frac{a}{\sin A} = \frac{b}{\sin B} \)
\( b = \frac{12 \times \sin 70°}{\sin 40°} = \frac{12 \times 0.940}{0.643} \approx 17.5 \)

Step 3: Find area:
\( \text{Area} = \frac{1}{2}ab \sin C \)
\( = \frac{1}{2} \times 12 \times 17.5 \times \sin 70° \)
\( = 6 \times 17.5 \times 0.940 \)
\( \approx 98.7 \) square units

⚡ Quick Summary

• Law of Sines relates sides to opposite angles
• Law of Cosines is like Pythagorean Theorem with an adjustment
• SSA case can have 0, 1, or 2 solutions (ambiguous case)
• Use Law of Cosines first for SAS and SSS
• Use Law of Sines first for AAS and ASA
• Sine area formula works for any triangle when you have SAS

📚 Decision Guide

⚠️ Common Mistakes to Avoid