Home›Lesson Plans›Mathematics›Trigonometry · Grade 12

The Law of Sines & Law of Cosines

Twelfth graders learn to solve any triangle — not just right triangles — using the Law of Sines and the Law of Cosines, choosing the right law based on which sides and angles are known.

Grade 12Triangles55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Use the Law of Sines.
  • ✓Use the Law of Cosines.
  • ✓Choose the right law.
  • ✓Solve any triangle.
Essential Question

Right-triangle trig only works on right triangles. But what about ALL the other triangles? Two powerful laws let you solve any triangle at all. When do you use each?

0
Lesson Phases
0
Vocabulary Terms
0
Standards Aligned
0
Interactive Task
Model · Interactive Tool

Which Law?

Project each item and have students choose the correct law. The tool explains when to use the Law of Sines vs. the Law of Cosines.

📐 Choose the right law for each triangleTry it
Choose the correct law for each case.
The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Beyond Right Triangles

5 min

Right-triangle trig fails on non-right triangles — what solves those?

👩‍🏫 Teacher Moves

  • Show a non-right triangle.
  • Ask how to solve it.
  • Set the goal: the two laws.

🎒 Student Actions

  • See the triangle.
  • Wonder how.
  • Predict a new tool.
2

I Do — Two Laws

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Law of Sines for AAS/ASA.
  • Law of Cosines for SSS/SAS.
  • Choose by what’s known.

🎒 Student Actions

  • Learn Sines.
  • Learn Cosines.
  • Choose the law.
3

We Do — Choose Together

13 min

Class uses Which Law?

👩‍🏫 Teacher Moves

  • Identify what’s given.
  • Choose the correct law.
  • Explain the choice.

🎒 Student Actions

  • Identify given.
  • Choose the law.
  • Explain it.
4

You Do — On Your Own

12 min

Students solve.

👩‍🏫 Teacher Moves

  • Solve triangles with the Law of Sines.
  • Solve with the Law of Cosines.
  • Choose correctly.

🎒 Student Actions

  • Solve with Sines.
  • Solve with Cosines.
  • Choose right.
5

Close — Close

5 min

One triangle.

👩‍🏫 Teacher Moves

  • Give a triangle.
  • Choose and apply the law.
  • Hand out the exit ticket.

🎒 Student Actions

  • Choose the law.
  • Apply it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School · Precalculus).

CCSS
G-SRT.11

Understand and apply the Law of Sines and the Law of Cosines to find measures in triangles.

CCSS
G-SRT.10

Prove the Laws of Sines and Cosines.

CCSS
G-SRT.9

Derive the formula for the area of a triangle.

CCSS
F-TF.8

Use trigonometric relationships.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Which-law flowchart.
  • Sentence frame: “Use the Law of ___ because ___.”
  • Label the given parts.

Support & Access

IEP / 504
  • Start with the Law of Sines.
  • Provide a formula sheet.
  • Identify given parts first.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Handle the ambiguous case (SSA).
  • Find a triangle’s area.
  • Solve real navigation problems.
Materials

What to gather

  • 📽️Projector / board
  • 📐Protractors
  • 💻Which Law?
  • 🧮Calculators
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

Law of Sinesa/sin A = b/sin B = c/sin CLaw of Cosinesc² = a² + b² − 2ab cos Coblique trianglea triangle with no right angleAASangle-angle-side givenSSSall three sides givenSASside-angle-side givenambiguous caseSSA, which may give two trianglesbearinga direction angle
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
When do you use the Law of Sines?
When you know an angle and its opposite side (like AAS or ASA cases).
QUESTION 2
When do you use the Law of Cosines?
For SSS or SAS cases (all sides, or two sides and the included angle).
QUESTION 3
What does the Law of Cosines become when the angle is 90°?
The Pythagorean theorem (c² = a² + b²).

Going deeper? The ambiguous case.

Have students explore the ambiguous (SSA) case of the Law of Sines, where the given information can produce zero, one, or two triangles. A printable triangle-laws sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Law of Sines & CosinesFlip
Card 1
Term
Tap to flip →
Meaning
0

Nice work!

Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Law of Sines & CosinesQuiz
Score: 0
1 / 6
Question 1
0%

Nice work!

Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Law of Sines & CosinesPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: AAS, ambiguous case, bearing, Law of Cosines, Law of Sines, oblique triangle, SAS, SSS
  1. SSA, which may give two triangles
  2. a/sin A = b/sin B = c/sin C
  3. c² = a² + b² − 2ab cos C
  4. angle-angle-side given
  5. side-angle-side given
  6. all three sides given
  7. a triangle with no right angle
  8. a direction angle

Part B · Show what you learned

  1. When do you use the Law of Sines?
  2. When do you use the Law of Cosines?
  3. What does the Law of Cosines become when the angle is 90°?
Answer key — Part A: 1) ambiguous case · 2) Law of Sines · 3) Law of Cosines · 4) AAS · 5) SAS · 6) SSS · 7) oblique triangle · 8) bearing
Part B: 1) When you know an angle and its opposite side (like AAS or ASA cases). 2) For SSS or SAS cases (all sides, or two sides and the included angle). 3) The Pythagorean theorem (c² = a² + b²).