The shape of the lesson
By the end of the lesson, learners can
- The three basic shapes: In the lab, drag the point round the unit circle and watch its height trace out the sine graph and its across-distance trace out the cosine graph.
- Stretches and translations: Every graph in Paper 1 is one of the three basic shapes changed by the transformations you already know.
- Reading the graph: Questions often give a sketch and ask you to find the constants, or ask how many solutions an equation has.
- Shape: sine starts at 0 going up, cosine starts at its maximum, tangent has asymptotes.
- Vertical: \(a\) sets the amplitude \(|a|\); \(c\) moves the centre line to \(y = c\).
- Horizontal: period \(\dfrac{2\pi}{b}\) (tan: \(\dfrac{\pi}{b}\)); \(x + k\) shifts left by \(k\).
- Key points: label max, min, intercepts and asymptotes with exact coordinates.
- Counting: solutions are crossings with a horizontal line inside the interval.
One picture every idea returns to
\(y = \sin x\) and \(y = \cos x\) have period \(2\pi\) (\(360^\circ\)) and range \(-1 \le y \le 1\); \(y = \tan x\) has period \(\pi\) (\(180^\circ\)) and asymptotes at \(x = \dfrac{\pi}{2} + n\pi\). For \(y = a\sin bx + c\): amplitude \(|a|\), period \(\dfrac{2\pi}{b}\), and the centre line moves to \(y = c\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Waves that repeat.. Sine and cosine are waves of height 1 that repeat every \(2\pi\); tangent repeats every \(\pi\) and shoots off to infinity. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: The three basic shapes | In the lab, drag the point round the unit circle and watch its height trace out the sine graph and its across-distance trace out the cosine graph. Project tab 1 of the lab (Trigonometric graphs lab, unit circle to graph) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Stretches and translations | Every graph in Paper 1 is one of the three basic shapes changed by the transformations you already know. Model the worked example Amplitude and period on the board, one line at a time. Your turn: State the range and period of \(y = 2 - 4\cos 3x\). | Pairs |
| 21 to 30 min | Teach and check: Reading the graph | Questions often give a sketch and ask you to find the constants, or ask how many solutions an equation has. Model the worked example Finding the constants on the board, one line at a time. Your turn: Find the set of values of \(k\) for which \(3\sin x + 2 = k\) has exactly two solutions for \(0 \le x \le \pi\). | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 22 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Reading the graph, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| The tangent graph repeats every \(\pi\), not every \(2\pi\), and it has no maximum or minimum. | Draw its asymptotes as dashed vertical lines and make the branches approach them without touching. |
| \(y = \sin(x + k)\) moves the graph LEFT by \(k\), not right; and in \(y = \sin 2x\) the period halves, it does not double. | Return to Stretches and translations: ask a pair to state the correct version and show it on the lab before moving on. |
| Watch the ends of the interval. | With \(0 \le x \le \pi\) the end points are included, so \(k = 2\) gives two solutions (\(x = 0\) and \(x = \pi\)); with \(0 < x < \pi\) it would give none. |
Graphs of trigonometric functions: worksheet
6 questions, 22 marks. Written in the style of the papers; no past paper question is reproduced.
(a) State the period and the range of \(y = 5\sin\dfrac{x}{2}\).
(b) State the period of \(y = 2\tan 3x\).
Sketch the graph of \(y = 1 - 2\cos x\) for \(0 \le x \le 2\pi\). State the coordinates of the maximum and minimum points and of the points where the graph crosses the axes.
For the graph of \(y = \tan\left(x - \dfrac{\pi}{6}\right)\), \(0 \le x \le \pi\), find the equation of the asymptote and the exact coordinates of the points where the graph crosses the axes.
The graph of \(y = a + b\cos 2x\) passes through the points \((0, 7)\) and \(\left(\dfrac{\pi}{2}, -1\right)\).
(a) Find \(a\) and \(b\).
(b) State the range of \(y\).
(c) Find the values of \(x\) in \(0 \le x \le \pi\) for which \(y = 3\).
Find the set of values of \(k\) for which the equation \(2 - 3\sin x = k\) has exactly two solutions for \(0 \le x \le \pi\).
The depth of water in a harbour \(t\) hours after midnight is modelled by \(d = 6 + 2.5\cos\left(\dfrac{\pi t}{6}\right)\) metres, for \(0 \le t \le 24\).
(a) State the greatest and least depths and the times at which they occur.
(b) A ship needs at least 7.25 m of water. For how many hours in the 24-hour period can it enter the harbour?
Answer key and marking notes
- (a) Period \(\dfrac{2\pi}{1/2} = 4\pi\); range \(-5 \le y \le 5\).
- (b) Period \(\dfrac{\pi}{3}\).
- Reflect \(y = \cos x\) in the \(x\)-axis, stretch by 2, move up 1: range \(-1 \le y \le 3\).
- Minima \((0, -1)\) and \((2\pi, -1)\); maximum \((\pi, 3)\). The \(y\)-intercept is \((0, -1)\).
- \(x\)-intercepts: \(1 - 2\cos x = 0\), \(\cos x = \tfrac{1}{2}\), so \(\left(\dfrac{\pi}{3}, 0\right)\) and \(\left(\dfrac{5\pi}{3}, 0\right)\).
- Asymptote: \(x - \dfrac{\pi}{6} = \dfrac{\pi}{2}\), so \(x = \dfrac{2\pi}{3}\).
- \(x\)-axis: \(x - \dfrac{\pi}{6} = 0\), so \(\left(\dfrac{\pi}{6}, 0\right)\).
- \(y\)-axis: \(y = \tan\left(-\dfrac{\pi}{6}\right) = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3}\), so \(\left(0, -\dfrac{\sqrt{3}}{3}\right)\).
- (a) At \(x = 0\): \(a + b = 7\). At \(x = \dfrac{\pi}{2}\): \(\cos\pi = -1\), so \(a - b = -1\). Hence \(a = 3\), \(b = 4\).
- (b) \(-1 \le y \le 7\)
- (c) \(3 + 4\cos 2x = 3\) gives \(\cos 2x = 0\); \(0 \le 2x \le 2\pi\), so \(2x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}\).
- \(x = \dfrac{\pi}{4}\) or \(x = \dfrac{3\pi}{4}\)
- For \(0 \le x \le \pi\), \(\sin x\) rises from 0 to 1 and back to 0, so \(y = 2 - 3\sin x\) falls from 2 to \(-1\) at \(x = \dfrac{\pi}{2}\) and rises back to 2.
- A horizontal line \(y = k\) meets this twice when \(-1 < k \le 2\); at \(k = -1\) it touches once.
- Answer: \(-1 < k \le 2\)
- (a) Period \(\dfrac{2\pi}{\pi/6} = 12\) hours. Greatest depth 8.5 m at \(t = 0, 12, 24\); least depth 3.5 m at \(t = 6, 18\).
- (b) \(6 + 2.5\cos\dfrac{\pi t}{6} \ge 7.25\) gives \(\cos\dfrac{\pi t}{6} \ge 0.5\).
- In one cycle \(0 \le \dfrac{\pi t}{6} \le 2\pi\): \(\dfrac{\pi t}{6} \le \dfrac{\pi}{3}\) or \(\dfrac{\pi t}{6} \ge \dfrac{5\pi}{3}\), i.e. \(t \le 2\) or \(t \ge 10\).
- Over 24 hours: \(0 \le t \le 2\), \(10 \le t \le 14\), \(22 \le t \le 24\).
- Total \(2 + 4 + 2 = 8\) hours
8 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the period of \(y = 2\cos\dfrac{x}{3}\), where \(x\) is in radians.
- A\(6\pi\)
- B\(\dfrac{2\pi}{3}\)
- C\(2\pi\)
- D\(3\pi\)
Find the range of \(y = 5 - 3\sin 2x\).
- A\(2 \le y \le 8\)
- B\(-3 \le y \le 3\)
- C\(3 \le y \le 7\)
- D\(5 \le y \le 8\)
The diagram shows the graph of a function for \(0 \le x \le 2\pi\). Which equation could it be?
- A\(y = 1 + 2\sin x\)
- B\(y = 1 + 2\cos x\)
- C\(y = 3\sin x\)
- D\(y = 1 + 3\sin x\)
Find the equation of the asymptote of \(y = \tan\left(x - \dfrac{\pi}{6}\right)\) for \(0 \le x \le \pi\).
- A\(x = \dfrac{2\pi}{3}\)
- B\(x = \dfrac{\pi}{3}\)
- C\(x = \dfrac{\pi}{2}\)
- D\(x = \dfrac{\pi}{6}\)
How many solutions does the equation \(4\cos 3x = 1\) have for \(0 \le x \le 2\pi\)?
- A6
- B2
- C3
- D9
The function \(f(x) = a + b\sin x\) satisfies \(f\left(\dfrac{\pi}{6}\right) = 5\) and \(f\left(\dfrac{3\pi}{2}\right) = -1\). Find \(a\) and \(b\).
- A\(a = 3,\ b = 4\)
- B\(a = 11,\ b = -12\)
- C\(a = 2,\ b = 3\)
- D\(a = 4,\ b = 2\)
Find the coordinates of the maximum point of \(y = 2\sin\left(x + \dfrac{\pi}{3}\right)\) for \(0 \le x \le 2\pi\).
- A\(\left(\dfrac{\pi}{6},\ 2\right)\)
- B\(\left(\dfrac{5\pi}{6},\ 2\right)\)
- C\(\left(\dfrac{\pi}{2},\ 2\right)\)
- D\(\left(\dfrac{7\pi}{6},\ -2\right)\)
The line \(y = k\) meets the curve \(y = 3\cos x\), for \(0 \le x \le 2\pi\), at exactly two points. Find the set of possible values of \(k\).
- A\(-3 < k \le 3\)
- B\(-3 < k < 3\)
- C\(-3 \le k \le 3\)
- D\(0 < k < 3\)
Quiz key
Here \(k = \tfrac{1}{3}\), so the period is \(2\pi \div \tfrac{1}{3} = 6\pi\): dividing \(x\) by 3 stretches the graph by factor 3 parallel to the \(x\)-axis. The tempting \(\dfrac{2\pi}{3}\) is the period of \(\cos 3x\), which squashes the graph instead. \(2\pi\) ignores the stretch (the 2 changes the amplitude, not the period), and \(3\pi\) uses half of \(2\pi\) times 3.
When \(\sin 2x = -1\), \(y = 5 + 3 = 8\); when \(\sin 2x = 1\), \(y = 5 - 3 = 2\). So \(2 \le y \le 8\). The tempting \(-3 \le y \le 3\) gives the range of \(3\sin 2x\) and forgets the vertical shift of 5. \(3 \le y \le 7\) uses the 2 from \(2x\) as if it were the amplitude, and \(5 \le y \le 8\) forgets that the graph goes below its centre line too.
The maximum is 3 and the minimum is \(-1\), so the centre line is \(y = 1\) and the amplitude is 2. The curve starts on the centre line and rises first, like \(\sin x\), so \(y = 1 + 2\sin x\). The tempting \(y = 1 + 2\cos x\) has the right max and min but would start at its maximum 3 when \(x = 0\). \(y = 3\sin x\) would go down to \(-3\), and \(y = 1 + 3\sin x\) would reach 4.
\(x - \dfrac{\pi}{6} = \dfrac{\pi}{2}\) gives \(x = \dfrac{2\pi}{3}\): the graph of \(\tan x\) is translated \(\dfrac{\pi}{6}\) to the right, so its asymptote moves right too. The tempting \(\dfrac{\pi}{3}\) moves it the wrong way (subtracting \(\dfrac{\pi}{6}\)). \(\dfrac{\pi}{2}\) ignores the translation, and \(\dfrac{\pi}{6}\) is where the curve crosses the \(x\)-axis.
The equation is \(\cos 3x = 0.25\). The period of \(\cos 3x\) is \(\dfrac{2\pi}{3}\), so there are 3 complete cycles in \([0, 2\pi]\), and the line \(y = 0.25\) cuts each cycle twice: 6 solutions (none at the ends, since \(\cos 0 = 1 \ne 0.25\)). The tempting 2 is the number for \(\cos x = 0.25\), forgetting the 3. 3 counts one per cycle, and 9 multiplies 3 cycles by 3.
\(\sin\dfrac{\pi}{6} = \tfrac{1}{2}\) and \(\sin\dfrac{3\pi}{2} = -1\), so \(a + \tfrac{1}{2}b = 5\) and \(a - b = -1\). Subtracting, \(\tfrac{3}{2}b = 6\), so \(b = 4\) and \(a = 3\). The tempting \(a = 2,\ b = 3\) treats 5 as the maximum (\(a + b = 5\)), but \(x = \dfrac{\pi}{6}\) is not where the maximum is. \(a = 11,\ b = -12\) takes \(\sin\dfrac{3\pi}{2} = 1\), and \(a = 4,\ b = 2\) fits only the first equation.
\(x + \dfrac{\pi}{3} = \dfrac{\pi}{2}\) gives \(x = \dfrac{\pi}{6}\), and the maximum value is 2, so \(\left(\dfrac{\pi}{6}, 2\right)\). The next maximum, at \(x + \dfrac{\pi}{3} = \dfrac{5\pi}{2}\), gives \(x = \dfrac{13\pi}{6}\), outside the interval. The tempting \(\dfrac{5\pi}{6}\) adds \(\dfrac{\pi}{3}\) instead of subtracting (adding inside the bracket moves the graph left). \(\dfrac{\pi}{2}\) ignores the translation, and \(\left(\dfrac{7\pi}{6}, -2\right)\) is the minimum point.
For \(-3 < k < 3\) the line crosses the cycle twice. At \(k = 3\) it touches the curve at both ends, \(x = 0\) and \(x = 2\pi\): two points, so \(k = 3\) is included. At \(k = -3\) it touches only at \(x = \pi\): one point, so \(k = -3\) is excluded. Answer \(-3 < k \le 3\). The tempting \(-3 < k < 3\) forgets that the interval includes both ends where \(\cos x = 1\). \(-3 \le k \le 3\) wrongly includes \(-3\), and \(0 < k < 3\) ignores the negative values.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.