Math lesson plan · 12th grade (Year 13 in England) · 60 minutes
Simplify trigonometric expressions with identities
Trigonometry
CCSS HSF.TF.C.8England: A-level Pure Mathematics
Objective and success criteria
Students will: Simplify trigonometric expressions with identities.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Use sin²θ + cos²θ = 1 to find exact values
Leads to: the next skill in this area.
Materials: Scientific calculators, Protractors and rulers, Right-triangle diagrams, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Use sin²θ + cos²θ = 1 to find exact values). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- θ is acute and sin θ = 20/29. Find cos θ as a fraction. → 21/29Prompt: sin²θ + cos²θ = 1.
- 90° < θ < 180° and sin θ = 20/29. Find cos θ as a fraction. → -21/29Prompt: Use sin²θ + cos²θ = 1 to find the size of cos θ.
- θ is acute and cos θ = 40/41. Find tan θ as a fraction in simplest form. → 9/40Prompt: Find sin θ first with the identity.
Hook · 6 min
- Show a ramp or a ladder against a wall and ask how steep it is.
- Share the objective: Simplify trigonometric expressions with identities. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Simplify: 1 − cos²x
- Step 1: sin²x
- Answer: sin²x
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Simplify: sin²x + cos²x + tan²x (0 / 1 / 1 + tan²x / sin x) → 1 + tan²xPrompt: Use sin²x + cos²x = 1 and tan x = sin x / cos x.
- Simplify: (sin x + cos x)² − 2 sin x cos x (1 / sin x / tan x / sin²x) → 1Prompt: Use sin²x + cos²x = 1 and tan x = sin x / cos x.
You do: independent practice · 17 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Simplify: cos x · tan x (cos x / 1 + tan²x / sin x / tan x) → sin xPrompt: Use sin²x + cos²x = 1 and tan x = sin x / cos x.
- Simplify: sin²x/(1 − sin²x) (1 / 0 / cos x / tan²x) → tan²xPrompt: Use sin²x + cos²x = 1 and tan x = sin x / cos x.
Watch out for
- Mixing up opposite and adjacent sides.
- Calculator in radians instead of degrees.
Support and stretch
Support
- Review first: Use sin²θ + cos²θ = 1 to find exact values.
- Use concrete materials: scientific calculators.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder