Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Use sin²θ + cos²θ = 1 to find exact values
Trigonometry
CCSS HSF.TF.C.8England: A-level Pure Mathematics
Objective and success criteria
Students will: Use sin²θ + cos²θ = 1 to find exact values.
- 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: Trigonometric identities; Decide the sign of sin, cos and tan in each quadrant
Leads to: Simplify trigonometric expressions with identities
Materials: Scientific calculators, Protractors and rulers, Right-triangle diagrams, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Trigonometric identities; Decide the sign of sin, cos and tan in each quadrant). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- θ is acute and cos θ = 4/5. Find sin θ. → 3/5Prompt: Use sin²θ + cos²θ = 1.
- Without a calculator, is cos 41° positive or negative? (Positive / Negative) → PositivePrompt: Which quadrant is the angle in?
- True for all angles θ? cos(−θ) = −cos θ (True / False) → FalsePrompt: Test it with a value, such as θ = 30°.
Hook · 3 min
- Show a ramp or a ladder against a wall and ask how steep it is.
- Share the objective: Use sin²θ + cos²θ = 1 to find exact values. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: θ is acute and sin θ = 8/17. Find cos θ as a fraction.
- Step 1: 1 − (8/17)² = 225/289
- Step 2: cos θ = 15/17
- Answer: 15/17
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- θ is acute and sin θ = 5/13. Find cos θ as a fraction. → 12/13Prompt: sin²θ + cos²θ = 1.
- 90° < θ < 180° and sin θ = 3/5. Find cos θ as a fraction. → -4/5Prompt: Use sin²θ + cos²θ = 1 to find the size of cos θ.
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- θ is acute and cos θ = 12/13. Find sin θ as a fraction. → 5/13Prompt: 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.
Watch out for
- Mixing up opposite and adjacent sides.
- Calculator in radians instead of degrees.
Support and stretch
Support
- Review first: Trigonometric identities.
- 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
- Move on to: Simplify trigonometric expressions with identities.
- 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