Law of Sines & Cosines

Solve any triangle — when Pythagorean theorem isn't enough

The Pythagorean theorem only works for right triangles. For any other triangle, you need the Law of Sines and the Law of Cosines.

The Law of Sines: a/sin A = b/sin B = c/sin C. The Law of Cosines: c² = a² + b² − 2ab·cos C.

In this lesson you'll see triangles drawn with segments and labels, solve for missing sides and angles, and explore the ambiguous case where two different triangles fit the same measurements.

Related Topics

Pythagorean TheoremIntro to TrigonometryThe Unit Circle

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What is the Law of Sines?
The Law of Sines: a/sin A = b/sin B = c/sin C. Each side divided by the sine of the opposite angle is equal. Use it when you know an angle and its opposite side.
What is the Law of Cosines?
The Law of Cosines: c² = a² + b² − 2ab·cos C. It reduces to the Pythagorean theorem when C = 90°. Use it for SAS or SSS configurations.
What is the ambiguous case?
The ambiguous case occurs with SSA. Two different triangles might satisfy the same measurements because sin B could yield two possible angles (B and 180° − B).
When do I use Law of Sines vs Law of Cosines?
Use Law of Sines for ASA, AAS, or SSA. Use Law of Cosines for SAS or SSS. SAS with Law of Cosines avoids the ambiguous case.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.