Welch Teaching & Coaching
Trigonometry • College Preparatory Mathematics

Unit 8: Law of Sines & Law of Cosines

Solve non-right triangles, handle the SSA ambiguous case, calculate area, and apply trigonometry to realistic geometry.

U8
Unit Road Map

Lesson Map

Use this table as your route through the unit. Lesson numbers jump directly to the matching explanation.

LessonTopicYou should be able to…
8.1Law of SinesUse angle-opposite-side pairs to solve AAS and ASA triangles.
8.2SSA & the Ambiguous CaseDetermine whether SSA information creates zero, one, or two triangles.
8.3Law of CosinesSolve SAS and SSS triangles.
8.4Area of an Oblique TriangleUse two sides and the included angle to calculate area.
8.5Problem SolvingChoose an appropriate law and interpret results in context.
Navigation

How the Unit Is Built

Each resource below is part of the same learning cycle. Select a card to jump to it.

Lesson MapSee the sequence and jump directly to any lesson.Content SummaryRead the ideas, methods, examples, and checks.Video LibraryOpen a targeted Khan Academy or YouTube search for each lesson.VocabularyReview the language needed to read and explain the mathematics.Sample ProblemsPractice with expandable worked solutions.Equations & RulesUse a compact reference for formulas and decision rules.Review GuidePrepare with a skill checklist and fresh mixed practice.Performance TaskApply the unit in a modeling or measurement context.
Learn

Content Summary

Work through these lessons in order. Focus on the meaning of the mathematics, the decision that selects a method, and a check that confirms the result.

Lesson 8.1

Law of Sines

Use angle-opposite-side pairs to solve AAS and ASA triangles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 8.2

SSA & the Ambiguous Case

Determine whether SSA information creates zero, one, or two triangles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 8.3

Law of Cosines

Solve SAS and SSS triangles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 8.4

Area of an Oblique Triangle

Use two sides and the included angle to calculate area.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 8.5

Problem Solving

Choose an appropriate law and interpret results in context.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Watch & Reinforce

Video Library

These links open focused video searches by lesson topic. Use them as reinforcement rather than as a substitute for showing your own reasoning.

Lesson 8.1

Law of Sines

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 8.2

SSA & the Ambiguous Case

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 8.3

Law of Cosines

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 8.4

Area of an Oblique Triangle

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 8.5

Problem Solving

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Language

Vocabulary

These terms should appear naturally in explanations, diagrams, and written reasoning.

oblique triangle
Write a definition in your own words and add an example or sketch in your STRIDES notes.
included angle
Write a definition in your own words and add an example or sketch in your STRIDES notes.
opposite side
Write a definition in your own words and add an example or sketch in your STRIDES notes.
Law of Sines
Write a definition in your own words and add an example or sketch in your STRIDES notes.
Law of Cosines
Write a definition in your own words and add an example or sketch in your STRIDES notes.
ambiguous case
Write a definition in your own words and add an example or sketch in your STRIDES notes.
SSA
Write a definition in your own words and add an example or sketch in your STRIDES notes.
SAS
Write a definition in your own words and add an example or sketch in your STRIDES notes.
SSS
Write a definition in your own words and add an example or sketch in your STRIDES notes.
AAS
Write a definition in your own words and add an example or sketch in your STRIDES notes.
ASA
Write a definition in your own words and add an example or sketch in your STRIDES notes.
Practice

Sample Problems

Try each problem before opening the solution. The examples are original practice aligned to the unit skills.

1

Law of Sines

A=42°, B=71°, and a=12. Find b.
Show solution
b/sin71°=12/sin42°, so b≈16.96.
2

Triangle sum

A=38° and B=64°. Find C.
Show solution
C=180−38−64=78°.
3

Law of Cosines

a=7, b=10, C=52°. Find c.
Show solution
c²=7²+10²−2(7)(10)cos52°, so c≈7.93.
4

SSS angle

a=8, b=11, c=13. Find angle C.
Show solution
cos C=(8²+11²−13²)/(2·8·11)=16/176; C≈84.8°.
5

Area

Two sides are 9 and 14 with included angle 37°. Find area.
Show solution
K=½(9)(14)sin37°≈37.9 square units.
6

SSA reasoning

Why can SSA produce two triangles?
Show solution
Sine has the same positive value for supplementary angles, so an inverse-sine angle may have a second possible angle that also fits the side data.
Reference

Equations & Rules

Know what each relationship means and when it applies; do not treat the box as a list of disconnected formulas.

Law of Sines
a/sin A = b/sin B = c/sin C
Law of Cosines
c²=a²+b²−2ab cos C
Area
K=½ab sin C
Triangle Sum
A+B+C=180°
SSA Check
An inverse-sine result may have a supplementary angle; test whether the remaining angles can form a triangle.
Prepare

Review Guide

Skill Checklist

  • I can use angle-opposite-side pairs to solve AAS and ASA triangles.
  • I can determine whether SSA information creates zero, one, or two triangles.
  • I can solve SAS and SSS triangles.
  • I can use two sides and the included angle to calculate area.
  • I can choose an appropriate law and interpret results in context.

How to review

  1. Complete the unit Skills Check packet without notes.
  2. Mark the exact skill behind each error.
  3. Return to that lesson and complete a fresh example.
  4. Explain the correction in words, then retry a similar problem.
  5. Finish with mixed practice so you must choose the method yourself.
Apply

Performance Task

Indirect Measurement Challenge

Determine a distance or height that cannot be measured directly. Build an oblique-triangle model, justify whether Law of Sines or Law of Cosines applies, and discuss measurement uncertainty.

Evidence to include

A labeled mathematical model, organized calculations, appropriate units and precision, a written interpretation, and a check or discussion of limitations.