The first lesson widens sine, cosine and tangent to every angle by turning a radius of length 1 about its centre. The signs in each quadrant and the reference angle fall out of the picture, and so does the symmetry that later lessons use.
Lessons 2 and 3 turn that into graphs: the wave of sine and cosine, its amplitude and period, and the shifts you met in Year 10 applied to a model of a tide and of a turning wheel. Lesson 4 solves equations such as sin x = 0.4 in an interval and finds every solution by symmetry.
The last two lessons go back to triangles that have no right angle. The sine rule is derived from two right-angled triangles, including the case where two different triangles fit; the cosine rule, the area formula and bearings finish the unit.
A graphic display calculator (GDC) is optional. Lessons 2 and 3 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Sine, cosine and tangent beyond 90 degrees Sine, cosine and tangent as coordinates on a circle of radius 1, signs in the four quadrants, tan as sin divided by cos, reference angles and the four angles that share one. Turn a radius through a full circle and watch the sign of sine, cosine and tangent change, then see the four angles that share a reference angle. No calculator
- Lesson 2 · Graphs of sine and cosine: amplitude and period The graphs of sine and cosine for 0 to 360 degrees, amplitude and period of y = a sin bx and y = a cos bx, the cosine as a shifted sine, and the link to Year 10 stretches and reflections. Change the amplitude and the number of waves in y = a sin bx, then slide along sine and cosine together to see that one is the other moved 90 degrees. GDC optional
- Lesson 3 · Shifts and periodic models The equation y = a sin b(x − c) + d, the midline and the highest and lowest values, horizontal shifts and the link to Year 10 translations, and writing and using a periodic model for tides and a wheel. Move a sine graph sideways and up and watch the first maximum and the midline change, then ride a wheel and see the seat's height as a graph of time. GDC optional
- Lesson 4 · Solving trigonometric equations in an interval Solving sin x = k, cos x = k and tan x = k for 0 to 360 degrees by finding the calculator angle and using symmetry, rearranging first, and counting solutions over more than one turn. Slide a horizontal line across a sine graph and a tangent graph and see every solution between 0 and 360 degrees appear. Scientific calculator
- Lesson 5 · The sine rule and the ambiguous case The sine rule from right-angled triangles, for a missing side or angle, the case of an obtuse angle, and the ambiguous case with the number of triangles. Drop a height in a triangle and see the sine rule appear, then swing a side like a compass arm to see when two triangles fit. Scientific calculator
- Lesson 6 · The cosine rule, area and bearings The cosine rule for a side and for an angle, the area formula one half ab sin C, deciding which rule to use, and bearings: the angle between two courses from a point and at a turn. See the cosine rule appear from a height in a triangle, then turn a ship's course and watch the distance between two ships change. Scientific calculator
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: angles, graphs and equations | A | After lesson 4 | 25 minutes |
| B1: The total of three sines | B | Mid-unit, after lesson 3 | 50 minutes |
| C1: Explain it to someone who missed it | C | After lesson 5 | 35 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 6 | 50 minutes |
| D1: Tide times at an island pier | D and C | End of unit, after lesson 6 | 50 minutes |
| D2: Two boats leave the harbour | D and C | End of unit, after lesson 6 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C: D1 models a tide with a cosine, D2 uses the sine and cosine rules and bearings for two boats. Set one, not both.
Task A1: Quick check: angles, graphs and equations (criterion A)
When: After lesson 4. Time: 25 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Show your working. Angles are in degrees. Question 1 is done without a calculator; use one for the rest and give angles to 1 decimal place unless a question says otherwise.
Exact values and symmetry
- Without a calculator, write down the exact value of (a) cos 300° (b) tan 135°. (2 marks)
- Given that sin 25° = 0.4226, write down the values of sin 155°, sin 205° and sin 335°. (3 marks)
Graphs
- For the graph of y = 3 cos 4x + 1, state the amplitude, the period and the least value of y. (3 marks)
- The graph of y = a sin bx, with a and b positive, has a greatest value of 5. It passes through the origin and next crosses the x-axis at x = 60°. Find a and b. (3 marks)
Equations in an interval
- Solve sin x = 0.35 for 0° ≤ x ≤ 360°. (3 marks)
- Solve 4 cos x + 1 = −1 for 0° ≤ x ≤ 360°. (3 marks)
- Solve 2 sin x = cos x for 0° ≤ x ≤ 360°. (3 marks)
- How many solutions does cos x = 0.8 have for 0° ≤ x ≤ 720°? Explain how you know without solving it. (2 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You answer some of the routine questions correctly, for example an exact value or the amplitude of a graph. |
| 3–4 | You answer most of the routine questions correctly, including a solution of an equation, and give at least one correct solution to the others. |
| 5–6 | You answer the routine questions correctly with clear methods, and find every solution to questions 5 to 7 in the interval. |
| 7–8 | You answer questions 1 to 7 correctly and explain your reasoning in question 8, including why each extra turn of the graph adds solutions. |
Task B1: The total of three sines (criterion B)
When: Mid-unit, after lesson 3. Time: 50 minutes. Calculator: scientific calculator. Criterion: B, investigating patterns.
Choose any angle x and look at the three angles x, x + 120° and x + 240°. Call the total of their sines T(x), so that
T(x) = sin x + sin (x + 120°) + sin (x + 240°).
This task is about what T(x) does as x changes, how far you can trust what you see, and why. Use a calculator in degree mode and give values to 3 decimal places.
Part 1. Collect values
- Work out T(x) for x = 0°, 20°, 50° and 75°. Record each of the three sines and the total in a table you design yourself.
- Choose two more values of x of your own, one of them larger than 90°, and add them to your table.
Part 2. Describe and predict
- Describe what you notice about T(x), in words and then as a general statement. Say what evidence you have.
- Use your statement to predict T(200°) and T(310°). Then work out both to check your predictions.
Part 3. Test and extend
- Does your statement still hold if the sines are replaced by cosines? Test it for at least three values of x.
- Investigate the total of the sines of (a) four angles x, x + 90°, x + 180° and x + 270° (b) three angles x, x + 100° and x + 200°. In each case say what changes and what stays the same.
- Choose a number of angles and a spacing of your own. Predict what the total of their sines will do, then test your prediction. State the conditions under which your general statement holds.
Part 4. Justify
- For the four angles in question 6(a), use what you know about sin (x + 180°) from lesson 1 to explain why your result holds for every x, not only the ones you tried.
- Explain, using the unit circle, why your result for x, x + 120° and x + 240° holds for every x. Say what a table of values can show and what it cannot.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you work out some values of T(x), notice something about them and make a prediction for T(200°). |
| 3–4 | You complete a table for question 1, suggest a general statement that fits your values and use it to predict T(200°) and T(310°). |
| 5–6 | You organize your values clearly, state a general rule that fits the sines and the cosines, and verify it with new values and with the changed angles in question 6. |
| 7–8 | You state a general rule that fits all your findings, say when it holds and when it does not, test it on a case you chose, and justify it with a reason that works for every x, not only the values you tried. |
Task C1: Explain it to someone who missed it (criterion C)
When: After lesson 5. Time: 35 minutes. Calculator: scientific calculator. Criterion: C, communicating.
Mei was away for lessons 4 and 5. She copied this working from a classmate, Tomas, and it has mistakes in it.
Tomas’s working
(a) Solve sin x = 0.3 for 0° ≤ x ≤ 360°.
sin x = 0.3 = 17.5°. The sine repeats every 180°, so the other solution is 17.5° + 180° = 197.5°.
(b) In triangle PQR, p = 8 cm, q = 10 cm and angle P = 40°. Find angle Q.
sin Q ÷ 10 = sin 40° ÷ 8, so sin Q = 0.803 and Q = 53.5°. There is one triangle.
Write a one-page explanation for Mei. She should be able to follow it on her own, without asking anyone.
Part 1. Put it right
- Find every mistake in Tomas’s working. For each one, say what he did, why it is wrong and write the correct working.
Part 2. Teach the idea
- Explain to Mei how to find all the solutions of an equation of the form sin x = k between 0° and 360°. Use a sketch of the sine graph or a unit circle diagram, for k = 0.3 and for k = −0.6.
- Explain to Mei how the sine rule can give two possible triangles when two sides and an angle that is not between them are known. Use Tomas’s triangle in a labelled diagram.
Part 3. Check your explanation
- Make up one triangle where the sine rule gives a second angle that is not possible, so that only one triangle fits. Say how a reader can tell.
- Read your page as if you were Mei. Check that you used the degree sign, an equals sign only between things that are equal, a labelled diagram, and the words solution, symmetry, sine rule and ambiguous.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Some of your explanation can be followed, but key steps or words are missing and the diagrams or notation are confusing. |
| 3–4 | You explain both parts using some correct words and notation, in a sensible order, so Mei could follow most of it. |
| 5–6 | Your steps are complete and ordered, with a labelled sketch or diagram, degree signs and mostly correct notation; Mei could follow it without help. |
| 7–8 | Every step is explained concisely, your diagrams and working match your words, notation is always correct, and the page is ordered so that Mei could learn from it alone. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 6. Time: 50 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Three sections, from familiar questions to ones you have not met in this form. Show your working everywhere: it earns credit even when an answer slips. Angles are in degrees; give lengths to 3 significant figures and angles to 1 decimal place unless a question says otherwise. Question 1 is done without a calculator.
Section 1. Familiar questions
- Write down the exact value of (a) sin 240° (b) cos 135°. (2 marks)
- For the graph of y = 2 sin 3x, state the amplitude and the period, and say how many complete waves there are from 0° to 360°. (3 marks)
- Solve 5 sin x = 2 for 0° ≤ x ≤ 360°. (3 marks)
- In triangle ABC, A = 48°, B = 71° and a = 9.2 cm. Find the length b. (3 marks)
Section 2. More complex questions
- A wheel at a night market has a height above the ground, in metres, of h = 9 − 8 cos 30t, where t is the time in minutes after the wheel starts. (a) Write down the least and the greatest height. (b) Write down the time for one complete turn. (c) Find the first time the height is 12 m. (6 marks)
- A triangle has sides 12 cm, 15 cm and 19 cm. Find its largest angle, and then its area. (4 marks)
Section 3. Unfamiliar questions
- In triangle XYZ, XY = 9 cm, YZ = 6 cm and angle X = 35°. Show that two different triangles fit this information, and find both possible values of angle Z. (4 marks)
- Given that cos 50° = 0.6428, find all the values of x between 0° and 360° for which sin x = 0.6428. Explain your reasoning. (3 marks)
- Two sides of a triangle are 10 cm and 14 cm and its area is 35 cm2. Find the two possible sizes of the angle between the sides. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You answer some Section 1 questions correctly, for example an exact value or the amplitude of a graph. |
| 3–4 | You answer most of Section 1 correctly and make progress in Section 2. |
| 5–6 | You answer Section 1 and most of Section 2 correctly, with clear methods, and make a start in Section 3. |
| 7–8 | You answer Sections 1 and 2 correctly and solve most of Section 3, explaining your reasoning. |
Task D1: Tide times at an island pier (criteria D and C)
When: End of unit, after lesson 6. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A ferry operator runs small boats between the mainland and a pier on an island in the Andaman Sea. At the end of the pier the depth of water, d metres, at t hours after high tide is modelled by
d = 2.6 + 1.2 cos 30t, where 30t is an angle in degrees and t is between 0 and 12.
A boat can dock only when the depth is at least 2.3 m. High tide on the day of this task is at 05:30.
Part 1. Understand the model
- State the greatest and the least depth the model gives, and the time between one high tide and the next. Say what each number in the model tells you about the tide.
- Find the depth of water 2 hours after high tide.
Part 2. Docking
- Find the times, to the nearest minute after high tide, between which boats cannot dock in the 12 hours after the first high tide. Write them as clock times on the day.
- Boats are due to arrive at 07:00, 08:30, 10:00, 11:30, 13:00, 14:30 and 16:00. Which of these can dock? Justify your answer.
Part 3. Does it make sense?
- The model uses exactly 12 hours between high tides. A real tide cycle is about 12 hours 25 minutes. How would this change the times in question 3, and the next unsafe window later in the day, and how far can you trust the timetable?
- Write a short note to the ferry operator with your recommendations, so that someone who has not seen your working can understand them.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You find some of the right numbers, for example the greatest and least depth or the depth at one time, but there is no decision about docking. |
| 3–4 | You find the depths, the time when docking is impossible and which arrivals can dock, and say whether your answers make sense. |
| 5–6 | You choose suitable methods for every part, reach correct answers, explain how accurate the times are and whether the timetable is sensible, and describe how the 12 hour model differs from a real tide. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, explain how accurate each time is and why, and justify whether your recommendation is reasonable for a real tide. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you reached the times or which arrivals can dock. |
| 3–4 | Your working for the times can be followed, with some correct notation, units and clock times. |
| 5–6 | Your working is clear and in order, usually with correct units, degree signs and clock times, and your note to the operator is mostly well organized. |
| 7–8 | Your working is clear, concise and well ordered, notation and units are always correct, and your note to the operator makes sense on its own. |
Task D2: Two boats leave the harbour (criteria D and C)
When: End of unit, after lesson 6. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
Two fishing boats leave a harbour H on the Andaman coast at the same time. Boat A sails at 24 km/h on a bearing of 070°. Boat B sails at 18 km/h on a bearing of 150°. Both keep a steady speed and course. Their radios work only when the boats are 15 km or less apart.
Part 1. After 30 minutes
- Draw a labelled diagram of the harbour and the two boats after 30 minutes, and find the angle AHB between their paths.
- Find the distance between the boats after 30 minutes.
- Find the bearing of boat B from boat A after 30 minutes, to the nearest degree.
Part 2. Losing contact
- After how many minutes do the boats lose radio contact? Show how you found it, and give your answer to the nearest minute.
Part 3. Does it make sense?
- The bearings are measured to the nearest degree. Find how much the distance in question 2 could change if the bearing of boat B were really 148° or 152°, and say what that means for your answer to question 4.
- Write a short note to the harbour radio operator, saying when to expect radio contact to end and how sure you are, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You find some of the right numbers, for example the angle AHB or a distance travelled, but there is no answer about radio contact. |
| 3–4 | You find the distance and the bearing after 30 minutes and a time for losing contact, and say whether your answers make sense. |
| 5–6 | You choose suitable methods throughout (angles, the cosine rule, the sine rule), reach correct answers, explain how accurate the time is and whether the answer is sensible in practice. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, explain how accurate your time is and why, and justify whether your recommendation is reasonable for real boats. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you reached the distance, the bearing or the time. |
| 3–4 | Your working for the distance and bearing can be followed, with some correct notation, units and a diagram. |
| 5–6 | Your working is clear and in order, with a labelled diagram, usually correct units and bearings written with three figures, and your note is mostly well organized. |
| 7–8 | Your working is clear, concise and well ordered, your diagram matches your calculations, notation and units are always correct, and your note makes sense on its own. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.