At a glance. One lesson of about 60 minutes. It assumes lessons 1 and 2 and Year 9 equation solving. Students solve sin x = k, cos x = k and tan x = k for 0 to 360 degrees, after rearranging, and count solutions over longer intervals. Quadratics in sin x are left for the Diploma.
Calculator: scientific calculator. A scientific calculator is needed for the inverse functions. Degree mode and the fact that the inverse key gives one angle are the two practical points.
What students will learn
Know
- The inverse key gives one angle.
- The other solutions follow from symmetry: 180 − p, 360 − p, p + 180.
- The solutions must lie in the stated interval.
Understand
- A horizontal line cuts a periodic graph more than once.
- Which rule applies depends on the function, not on the sign of k.
- sin x ÷ cos x = tan x turns a ratio into one equation.
Do
- Solve sin x = k, cos x = k and tan x = k in 0 to 360 degrees.
- Rearrange before solving.
- Count solutions over more than one turn.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the symmetry rules and the interval (check and questions 1 to 8).
- Criterion B. Investigating patterns: the figures. Ask students to predict how many solutions k = 0 will have before it appears.
- Criterion C. Communicating: writing the calculator angle, the symmetry step and the final list in separate lines.
- Criterion D. Applying mathematics in real-life contexts: the seat on the wheel, passing a height twice per turn.
The one thing to do with the figure
Predict the crossings. In the first figure, ask “Where will the second solution be for k = 0.2?” before the line moves, and ask students to say its distance from 90°.
In the second figure, ask why the second solution is exactly 180° from the first.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the seat on the wheel. Collect predictions of how many times. |
| 10 min | First figure and key idea: the symmetry rules. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask where the cosine is positive. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | First and second; 180°; x = 5. |
|---|---|
| Hook | Twice, once going up and once coming down. If the first is at 40° the second is at 180° − 40° = 140°, by the symmetry of the sine. |
| Worked examples | 44.4° and 135.6°; 101.5° and 258.5°. |
| Faded example | 63.4° and 243.4°. |
| 1. A complete solution set | B. A uses the sine pattern; C is the third quadrant; D stops at the calculator. |
| 2. The second solution of a sine equation | 121.8. 180 − 58.2. |
| 3. The larger solution of a tangent equation | 230.2. 50.2 + 180. |
| 4. A negative value of the sine | 194.5. 180 + 14.5. |
| 5. An exact equation | 240. 360 − 120. |
| 6. A ratio of sine and cosine | 213.7. 33.7 + 180. |
| 7. Solutions over one and a half turns | 3. 75.5°, 284.5° and 435.5°. |
| 8. Rearranging first | A. B solves tan x = 1.4; C has the wrong second angle; D mixes the signs. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 (the check) | Uses the sine pattern (180 − p), uses the third quadrant, or stops at one answer. | Ask: where is the cosine positive? |
| 301.8, 238.2 or 31.8 in question 2 | Uses the cosine pattern (360 − p), adds 180, or uses the complement. | Sketch the sine curve and mark the mirror image about 90°. |
| 129.8, 309.8 or 50.2 in question 3 | Uses sine or cosine symmetry, or gives only the first. | Say: tangent repeats every 180°. |
| −14.5, 165.5 or 345.5 in question 4 | Gives the raw calculator angle, uses the positive rule, or the larger answer. | Ask which quadrants have a negative sine. |
| 120, 300 or 210 in question 5 | Gives the smaller solution, uses 360 − 60 where the cosine is positive, or uses the reference angle 30°. | Find the reference angle from the exact value first, then the quadrants with a negative cosine. |
| 146.3, 33.7 or 326.3 in question 6 | Uses the sine pattern, gives the first solution, or uses the cosine pattern. | Rewrite as tan x first. |
| 2, 1 or 4 in question 7 | Counts one turn only, the calculator's single angle, or two full turns. | Sketch the curve as far as 540° and draw the line. |
| B, C or D in question 8 | Solves tan x = 1.4, or gives angles that are not 180° apart. | Rearrange first, then check the pair differ by 180°. |
Questions worth asking
- Why does a horizontal line cut a sine graph twice in one turn? The graph rises to a peak and falls again, passing each level between the minimum and maximum twice.
- Why is the second solution of cos x = k at 360 − p? The cosine is the sideways distance, and the circle is symmetric in the horizontal axis.
- Why can sin x = 2 not be solved? The sine is never more than 1.
- How does the unit circle give the same answer as the graph? The tip of the radius at p and at 180 − p has the same height.
Differentiation
A way in for anyone stuck
- Give a printed sine graph for 0 to 360 and a ruler, so students can draw the line and read the crossings.
- Use a three-column frame: the calculator angle, the symmetry rule, the list.
- Provide a table of the rules for sine, cosine and tangent.
Stretch
- Solve 2 sin x = 3 cos x for 0 to 360 degrees, then explain why dividing by cos x is safe here.
- Solve sin x = cos x for 0 to 360 degrees and explain the answers with the unit circle.
Students learning in English as an additional language
“Solution” is a value that makes the equation true. “Interval” is the set of values between two ends, here 0 and 360 degrees. “Symmetry” means a graph looks the same after a reflection. “Inverse” is the operation that undoes another.
| Word | Plain meaning | Example |
|---|---|---|
| solution | a value of the unknown that makes the equation true | an angle that gives the right sine |
| interval | all the values between two end values | 0° to 360° |
| symmetry | a graph that looks the same after a reflection | the sine graph about 90° |
| inverse | the operation that undoes another | sin⁻¹ undoes sin |
| principal value | the one angle that the calculator's inverse key gives | −30° for sin⁻¹ (−0.5) |
Sentence starters for explaining aloud or in writing:
- The calculator gives …, which is in the … quadrant.
- The other solution is … because ….
- In 0° to 360° there are … solutions.
Two things not to say
“The calculator gives the answer.” It gives one of the answers.
“Always subtract from 180.” That rule belongs to sine only.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- RearrangingWrite sin x = k from 4 sin x = 3.
sin x = 0.75. - Symmetrysin x = 0.3 has a solution at 17.5°. Find the other solution in 0° to 360°.
162.5°. 180 − 17.5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA seat's height is h = 10 + 8 sin x metres, x in degrees. At what angles between 0° and 360° is h = 14?
x = 30° and 150°. sin x = 0.5. - Earlier work (Lesson 3)State the midline of y = 4 + 3 cos x.
y = 4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningDoes cos x = 0.5 have a solution in 90° to 270°? Explain.
No: cosine is positive only in the first and fourth quadrants. - Working backwardsThe solutions of an equation in 0° to 360° are 30° and 330°. Which function and value?
cos x = 0.866, that is √3 ÷ 2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a printed graph of y = sin x and a ruler and ask for the angles at heights 0.2, 0.5 and −0.5.