Law of Sines vs Cosines | Trigonix Blog

Choosing between the Law of Sines and Law of Cosines is crucial for solving oblique triangles efficiently. This guide provides clear decision-making principles with practical examples.

Law of Sines

a/sinA = b/sinB = c/sinC

When to use:

When you know two angles and any side (AAS or ASA)
When you know two sides and a non-included angle (SSA)

Law of Cosines

c² = a² + b² - 2ab·cosC

When to use:

When you know all three sides (SSS)
When you know two sides and the included angle (SAS)

Practical Example

Scenario: Triangle with sides a=7, b=10, and angle C=40°

Solution: This is SAS case → Use Law of Cosines first:

c² = 7² + 10² - 2(7)(10)cos40° ≈ 49 + 100 - 107.28 ≈ 41.72

c ≈ √41.72 ≈ 6.46

Now use Law of Sines to find other angles

Decision Flowchart

Count known elements (angles/sides)
Identify known combination (AAS, ASA, SSA, SAS, SSS)
Apply appropriate law based on combination
Use the other law to find remaining elements if needed

Pro Tip: The Law of Cosines is generally more computationally intensive but works in all cases. When possible, use the Law of Sines first for simpler calculations.

Common Mistakes

Using Law of Sines for SSS cases
Applying Law of Cosines to right triangles unnecessarily
Forgetting to consider the ambiguous case (SSA) with Law of Sines

Mastering these laws enables you to solve any oblique triangle, essential for navigation, surveying, and physics applications.

🛠 When to Use the Law of Sines vs Cosines

Law of Sines

Law of Cosines

Practical Example

Decision Flowchart

Common Mistakes