Law of sines and law of cosines – Grade 11 math lesson
Lesson 14 of 23 in the Grade 11 math course
US version. US grades, US spelling and dollars ($). · UK Maths Department
A complete 45-minute lesson for grade 11 (ages 16–17) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.
- Time
- 45 minutes + 20 min homework
- Grade
- Grade 11 (ages 16–17)
- You need
- Paper, a pencil and this page
- Graded
- Instantly, with explanations
For parents: how to use this lesson
You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is graded straight away with a worked explanation, so you never have to grade anything.
- Read what your child will learn and the big ideas together.
- Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
- Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
- Finish with the exit ticket and the talk question: “How do you decide which rule to use?”
- Later that day or the next day: the homework (20 minutes), then a practice test.
Watch out for
- Using the cosine rule when a side–opposite-angle pair is known. A pair → sine rule; two sides and the angle between → cosine rule.
- Pressing the calculator in the wrong order for the cosine rule. Work out 2bc cos A first, then subtract.
Tip for home: Work out a distance across a lake or field from two angles.
What your child will learn
Learning goal: use the law of sines, the law of cosines and ½ab sin C in any triangle.
Success looks like
- I can use the sine rule.
- I can use the cosine rule.
- I can find the area of any triangle.
Key words
- law of sines
- links sides with their opposite angles
- law of cosines
- like Pythagoras for any triangle
- included angle
- the angle between two given sides
- opposite
- across the triangle from the angle
What to have ready
- Paper and a pencil
- This page on a computer, tablet or phone
- For the hands-on part: Draw a triangle accurately, measure it, then check the rules with your measurements.
The big ideas (for the adult)
The law of sines: a/sin A = b/sin B = c/sin C. Use it when you know a side and its opposite angle.
The law of cosines: a² = b² + c² − 2bc cos A. Use it with two sides and the angle between them, or three sides.
The area of any triangle is ½ab sin C, using two sides and the angle between them.
The lesson, step by step
| Part | Time | How it is checked |
|---|---|---|
| Starter: quick recall | 5 min | Graded by Classfern |
| Teaching input | 10 min | Not assessed, self-checked |
| Worked examples and guided practice | 9 min | Not assessed, graded by Classfern |
| Independent practice | 15 min | Graded by Classfern |
| Closure and exit ticket | 6 min | Graded by Classfern, self-checked |
Starter, "your turn", practice and exit ticket are graded by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.
The lesson
It runs here on the page. No sign-up and no names – progress is kept only in this browser.
Homework and practice at home
20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all graded instantly.
More practice
More Grade 11 math lessons
- Complex numbers
- Radicals
- Vectors
- Laws of exponents
- Square roots and cube roots
- Systems of equations
- Rational expressions and polynomials
- Arithmetic and geometric sequences
- Iteration
- Functions
- Graphing functions
- Exponential functions
- Trigonometric functions
- Coordinate geometry
- Counting outcomes
- Experimental probability
- Compound events
- Conditional probability
- Expected value
- Box plots and interquartile range
- Scatter plots and lines of fit
- Sampling and making inferences
Curriculum
Common Core State Standards HSG.SRT.D.11: “(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).”
Common Core State Standards HSG.SRT.D.9: “(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.”
Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.