---
title: Law of Sines & Cosines
description: The Law of Sines and the Law of Cosines relate the sides and angles of non-right triangles. Learners will understand how to solve for unknown distances and angles in oblique triangles, which is essential for navigation, surveying, and physics.
category: mathematics
subcategory: geometry-and-trigonometry
difficulty: beginner, intermediate, advanced
url: /subject/law-of-sines-and-cosines
---

# Law of Sines & Cosines

The Law of Sines and the Law of Cosines relate the sides and angles of non-right triangles. Learners will understand how to solve for unknown distances and angles in oblique triangles, which is essential for navigation, surveying, and physics.

## Available Resources

1 Courses • 4 Websites

## Courses

### 1. Trigonometry Khan Academy

Master law of sines & cosines with Sal Khan's free Trigonometry course! Ace triangles & trigonometry calculations.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/trigonometry

**Tags:** trigonometry, law-of-sines, law-of-cosines, unit-circle

## Websites

### 1. The Laws of Sines and Cosines

**Author:** Alexander Bogomolny

Reference page collecting classical proofs of both laws: Euclid's Elements II.12 treatment of the obtuse case, two proofs without words from Nelsen using power of a point and Ptolemy's theorem, Maor's projection argument, and dissection proofs.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www.cut-the-knot.org/pythagoras/cosine2.shtml

**Tags:** law-of-sines, law-of-cosines, geometric-proof, euclid

### 2. Precalculus (Stitz-Zeager), Section 11.3: The Law of Cosines

**Author:** Carl Stitz and Jeff Zeager

Companion section deriving the Law of Cosines by placing a triangle at the origin and applying the distance formula, covering the SAS and SSS cases and continuing to Heron's formula. Twenty-seven applied exercises with complete answers.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/11:_Applications_of_Trigonometry/11.03:_The_Law_of_Cosines

**Tags:** law of cosines, trigonometry, oblique triangles, heron's formula, precalculus

### 3. Precalculus (Stitz-Zeager), Section 11.2: The Law of Sines

**Author:** Carl Stitz and Jeff Zeager

Open-textbook section proving the Law of Sines separately for acute, obtuse, and right triangles, then systematizing the ambiguous SSA case into zero, one, or two triangles. Includes bearing and navigation applications and 38 answered exercises.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/11:_Applications_of_Trigonometry/11.02:_The_Law_of_Sines

**Tags:** law of sines, trigonometry, oblique triangles, ambiguous case, precalculus

### 4. mathsisfun.com

MathIsFun.com is a beginner-friendly math site that explains concepts with clear, step-by-step lessons and interactive activities. It covers topics from algebra and linear equations to geometry, fractions, data, and more, with practice problems, worksheets, and quizzes.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathsisfun.com

**Tags:** websites, mathematics-statistics, algebra

## Youtubes

### 1. Classic Math Proofs of the Law of Sines and the Law of Cosines

**Author:** Steve Chow

Eleven-minute blackboard derivation of both laws in one sitting, dropping altitudes in an oblique triangle and handling the obtuse configuration rather than asserting the formulas. Builds the geometric reason the ratios hold before any problem-solving.

**Difficulty:** Beginner | **Language:** English | **Price:** Free

**Link:** https://www.youtube.com/watch?v=843Uw-Hj-OQ

**Tags:** law-of-sines, law-of-cosines, trigonometry-proofs, oblique-triangles

---

*This content is part of Dantes.io - Your Treasure Map to Knowledge*

*Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).*

View this page online: https://dantes.io/subject/law-of-sines-and-cosines