Non-right Triangles: Law of Sines

Section 8.1 | Lesson 1 of 8

Learning Objectives:

• Use the Law of Sines to solve oblique triangles
• Solve ASA and AAS triangle problems
• Recognize and handle the ambiguous case (SSA)
• Find the area of an oblique triangle

Why Do We Need New Tools?

Right triangles: SOH-CAH-TOA works perfectly

Oblique triangles: No right angle = no hypotenuse

• SOH-CAH-TOA requires a right angle
• Most real-world triangles aren't right triangles
• We need a relationship that works for ANY triangle

What is an Oblique Triangle?

An oblique triangle is any triangle that is NOT a right triangle.

To "solve" an oblique triangle means finding:

• All three angles (α, β, γ)
• All three sides (a, b, c)

Requirement: At least 3 pieces of information, including at least one side.

Standard Triangle Labeling

Convention: Each angle is opposite its matching side.

• Angle α (alpha) is opposite side a
• Angle β (beta) is opposite side b
• Angle γ (gamma) is opposite side c

Deriving the Law of Sines

Drop an altitude h from angle β to side b:

In the left triangle: →

In the right triangle: →

The Law of Sines

Since both expressions equal h:

Rearranging gives us:

The same process with other altitudes proves:

Using the Law of Sines

To use this law, you need:

• At least one complete opposite pair (a side AND its opposite angle)
• Plus one additional piece of information

Works for: ASA, AAS, and SSA cases

Case 1: ASA (Angle-Side-Angle)

Given: Two angles and the side between them.

Strategy:

1. Find the third angle: α + β + γ = 180°
2. Use Law of Sines twice to find both missing sides

Example: ASA Triangle

Problem: In triangle ABC, α = 35°, γ = 85°, and b = 12. Solve the triangle.

Step 1: Find β

Step 2: Find side a

Example: ASA (continued)

Step 3: Find side c

Complete Solution:

• α = 35°, β = 60°, γ = 85°
• a ≈ 7.95, b = 12, c ≈ 13.81

Case 2: AAS (Angle-Angle-Side)

Given: Two angles and a non-included side.

Strategy: Same as ASA!

1. Find the third angle: α + β + γ = 180°
2. Use Law of Sines twice to find both missing sides

Key insight: Once you know all three angles, it doesn't matter which side you started with.

Practice: ASA and AAS

Solve these triangles. Round to one decimal place.

1. ASA: α = 40°, β = 75°, c = 15
2. AAS: β = 50°, γ = 70°, a = 8

Pause and solve.

Answers

Problem 1: γ = 65°, a ≈ 10.6, b ≈ 16.0

Problem 2: α = 60°, b ≈ 7.1, c ≈ 8.7

Verification tip: The largest angle is always opposite the largest side.

The Ambiguous Case: SSA

Given: Two sides and an angle opposite one of them.

The problem: This may yield:

• No solution (impossible triangle)
• One solution (unique triangle)
• Two different solutions (two valid triangles!)

Why SSA is Ambiguous

When given sides a and b with angle α opposite side a:

The side a can "swing" to different positions.

Think of it like a door:

• Side b is the door frame
• Side a is the door
• Where does the door reach the floor?

SSA: The Decision Process

1. Use Law of Sines to find sin of the unknown angle
2. If sin θ > 1: No solution (impossible)
3. If sin θ ≤ 1: Calculate θ = sin⁻¹(value)
4. Check both possibilities:
   • θ (from calculator)
   • 180° − θ (the supplement)
5. Verify each leads to a valid triangle (angles sum to 180°)

Example: SSA with Two Solutions

Problem: Given a = 10, b = 12, α = 40°. Solve the triangle.

Step 1: Find sin β

Step 2: Two possible angles

Visualizing the Two Solutions

From point B, side a = 10 swings as an arc and intersects the base at two points: C₁ and C₂.

Both triangles ABC₁ and ABC₂ satisfy a = 10, b = 12, α = 40°!

Example: SSA Two Solutions (continued)

Check both:

Solution 1: β₁ = 50.5°

• γ₁ = 180° − 40° − 50.5° = 89.5° ✓ Valid!

Solution 2: β₂ = 129.5°

• γ₂ = 180° − 40° − 129.5° = 10.5° ✓ Valid!

Both triangles exist! Use Law of Sines to find c for each.

Example: SSA with No Solution

Problem: Given a = 5, b = 12, α = 60°. Solve the triangle.

Since sin β > 1: No solution exists!

The side a = 5 is too short to form a triangle.

Practice: SSA Cases

Determine how many triangles exist and solve if possible.

1. a = 8, b = 10, α = 50°
2. a = 15, b = 10, α = 40°

Pause and solve.

Answers: SSA Practice

Problem 1: Two solutions

• Solution 1: β ≈ 73.7°, γ ≈ 56.3°, c ≈ 8.7
• Solution 2: β ≈ 106.3°, γ ≈ 23.7°, c ≈ 4.2

Problem 2: One solution

• β ≈ 25.4°, γ ≈ 114.6°, c ≈ 21.2
• (The supplement 154.6° would make angles exceed 180°)

Area of an Oblique Triangle

Standard formula: Area = ½ × base × height

Problem: We often don't know the height directly.

Solution: Express height using sine!

If we know sides a and b with included angle γ:

The Area Formula

Also equivalent:

Use whichever pair you know!

Example: Finding Area

Problem: Find the area of a triangle with sides a = 8, b = 11, and included angle γ = 70°.

Application: Surveying

Problem: A surveyor needs to find the distance across a lake. From point A, she measures 85° to point B. She walks 500m to point C and measures 62° to point B.

1. Find angle B: 180° − 85° − 62° = 33°
2. Apply Law of Sines:

Summary: Key Takeaways

• Law of Sines:
• ASA/AAS: Find third angle first, then Law of Sines (unique solution)
• SSA: Check for 0, 1, or 2 solutions (ambiguous case)
• Area: (need two sides and included angle)

Common Mistakes to Avoid

• Using Law of Sines for SAS: Use Law of Cosines instead
• Forgetting the ambiguous case: Always check both angles in SSA
• Calculator mode errors: Ensure degrees mode, not radians
• Inverse sine limitation: sin⁻¹ gives angles in [−90°, 90°]; obtuse angles need 180° − θ
• Rounding too early: Keep extra decimals until the final answer

Next Steps

You've mastered:

• Law of Sines for ASA, AAS, and SSA
• Area formula for oblique triangles

Coming Up (Section 8.2):

• Law of Cosines for SAS and SSS cases
• When to use each law

Practice: Complete the assigned problems from Section 8.1