Math lesson plan · 11th grade (Year 12 in England) · 45 minutes
Use reference angles for angles beyond 90°
Trigonometry
CCSS HSF.TF.A.2England: A-level Pure Mathematics
Objective and success criteria
Students will: Use reference angles for angles beyond 90°.
- 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: Decide the sign of sin, cos and tan in each quadrant
Leads to: Solve trigonometric equations between 0° and 360°
Materials: Scientific calculators, Protractors and rulers, Right-triangle diagrams, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (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.
- Without a calculator, is cos 93° positive or negative? (Positive / Negative) → NegativePrompt: Which quadrant is the angle in?
- θ is in the third quadrant. Is tan θ positive or negative? (Positive / Negative) → PositivePrompt: cos is the x-coordinate and sin is the y-coordinate on the unit circle.
- Without a calculator, is sin 112° positive or negative? (Positive / Negative) → PositivePrompt: Which quadrant is the angle in?
Hook · 5 min
- Show a ramp or a ladder against a wall and ask how steep it is.
- Share the objective: Use reference angles for angles beyond 90°. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: What is the reference angle for 320°?
- Step 1: 360° − 320° = 40°
- Answer: 40
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- What is the reference angle for 184°? → 4Prompt: The reference angle is the acute angle between the terminal side and the x-axis.
- Find the exact value of cos 150°. (−√3/2 / √3/2 / −1/2 / √2/2) → −√3/2Prompt: Find the reference angle (the acute angle to the x-axis).
You do: independent practice · 10 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 · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- What is the reference angle for 142°? → 38Prompt: The reference angle is the acute angle between the terminal side and the x-axis.
- Find the exact value of cos 135°. (√2/2 / 1/2 / √3/2 / −√2/2) → −√2/2Prompt: Find the reference angle (the acute angle to the x-axis).
Watch out for
- Mixing up opposite and adjacent sides.
- Calculator in radians instead of degrees.
Support and stretch
Support
- Review first: Decide the sign of sin, cos and tan in each quadrant.
- 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: Solve trigonometric equations between 0° and 360°.
- 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-11.
Free lesson plan from eurekami.com/lesson-plan-builder