The shape of the lesson
By the end of the lesson, learners can
- Sine and cosine waves: In the lab, send the point round the unit circle and watch its height draw the sine graph and its horizontal position draw the cosine graph.
- The tangent graph: \(\tan x = \dfrac{\sin x}{\cos x}\), so wherever \(\cos x = 0\) the tangent is undefined.
- Symmetry gives the second angle: A calculator gives only one angle for each value.
- Sine: starts at 0, peaks at \((90^\circ, 1)\), lowest at \((270^\circ, -1)\).
- Cosine: starts at 1, lowest at \((180^\circ, -1)\), back to 1 at \(360^\circ\).
- Tangent: zeros at \(0^\circ, 180^\circ, 360^\circ\); asymptotes at \(90^\circ, 270^\circ\); period \(180^\circ\).
- Second angle: \(180^\circ - \alpha\) (sin), \(360^\circ - \alpha\) (cos), \(180^\circ + \alpha\) (tan).
- Count solutions: draw the horizontal line and count the crossings.
One picture every idea returns to
\(y = \sin x\) and \(y = \cos x\) are waves between \(-1\) and \(1\) that repeat every \(360^\circ\). \(y = \tan x\) repeats every \(180^\circ\) and has vertical asymptotes at \(90^\circ\) and \(270^\circ\). Their symmetry gives a second angle with the same value: \(\sin(180^\circ - x) = \sin x\), \(\cos(360^\circ - x) = \cos x\), \(\tan(180^\circ + x) = \tan x\).
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, Angles that go round and round.. Extended. 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: Sine and cosine waves | In the lab, send the point round the unit circle and watch its height draw the sine graph and its horizontal position draw 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. Model the worked example A table of values on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: The tangent graph | \(\tan x = \dfrac{\sin x}{\cos x}\), so wherever \(\cos x = 0\) the tangent is undefined. Model the worked example Using the period of tan on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Symmetry gives the second angle | A calculator gives only one angle for each value. Model the worked example Same sine on the board, one line at a time. Your turn: Given that \(\sin 38^\circ = 0.616\), find the two angles between \(0^\circ\) and \(360^\circ\) with \(\sin x = -0.616\). | Jigsaw |
| 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, 17 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 Symmetry gives the second angle, 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.
Jigsaw
Split the content into expert pieces. Expert groups master one piece each, then regroup so every home team has one expert per piece, and each expert teaches their piece to the team.
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 |
|---|---|
| Sketching sine and cosine with sharp corners at the top and bottom, or as straight zig-zags. | The graphs are smooth curves that turn gently at their maximum and minimum. |
| Drawing the tangent curve across the asymptote at \(90^\circ\). | Each branch climbs up towards \(+\infty\) on the left of the asymptote and comes back from \(-\infty\) on the right. |
| Using the sine pattern \(180^\circ - \alpha\) for cosine. | For cosine the partner of \(70^\circ\) is \(290^\circ\) (\(360^\circ - 70^\circ\)), not \(110^\circ\). Always check with a quick sketch. |
Graphs of sine, cosine and tangent: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
For the graph of \(y = \cos x\), \(0^\circ \le x \le 360^\circ\), write down (a) the period (b) the coordinates of the minimum point (c) the values of \(x\) where the graph crosses the \(x\)-axis.
Sketch the graph of \(y = \tan x\) for \(0^\circ \le x \le 360^\circ\). Mark the asymptotes and the points where the graph crosses the \(x\)-axis.
Given that \(\sin 38^\circ = 0.616\), find the two values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\sin x = -0.616\).
Without solving them, state how many solutions each equation has for \(0^\circ \le x \le 360^\circ\). Give a reason each time.
(a) \(\cos x = 0.3\) (b) \(\sin x = 1.2\) (c) \(\sin x = 1\)
Given that \(\tan 72^\circ = 3.08\), find the other value of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\tan x = 3.08\), and one value for which \(\tan x = -3.08\).
A seat on a big wheel is \(h\) metres above the ground after the wheel has turned through \(x^\circ\), where \(h = 10 - 8\cos x\).
(a) Find the height of the seat when \(x = 0^\circ\).
(b) Use your knowledge of the cosine graph to find the greatest height of the seat and the value of \(x\) at which it occurs.
(c) Find the two values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(h = 14\).
Answer key and marking notes
- (a) \(360^\circ\)
- (b) \((180^\circ, -1)\)
- (c) \(x = 90^\circ\) and \(x = 270^\circ\)
- Crosses the \(x\)-axis at \((0^\circ, 0)\), \((180^\circ, 0)\) and \((360^\circ, 0)\).
- Vertical asymptotes at \(x = 90^\circ\) and \(x = 270^\circ\).
- From \(0^\circ\) the curve rises through \((45^\circ, 1)\) towards the asymptote at \(90^\circ\); it reappears from far below, rises through \((180^\circ, 0)\) and \((225^\circ, 1)\) towards \(270^\circ\); then rises from far below to \((360^\circ, 0)\).
- Sine is negative for \(180^\circ < x < 360^\circ\).
- \(x = 180^\circ + 38^\circ = 218^\circ\) and \(x = 360^\circ - 38^\circ = 322^\circ\)
- (a) 2: the line \(y = 0.3\) crosses the cosine wave once before \(90^\circ\) and once after \(270^\circ\).
- (b) 0: \(\sin x\) is never greater than 1.
- (c) 1: \(y = 1\) touches the sine graph only at its maximum, \(x = 90^\circ\).
- Period \(180^\circ\): \(72^\circ + 180^\circ = 252^\circ\).
- \(\tan x\) is negative between \(90^\circ\) and \(180^\circ\): \(180^\circ - 72^\circ = 108^\circ\) (or \(360^\circ - 72^\circ = 288^\circ\)).
- (a) \(h = 10 - 8 \times 1 = 2\) m
- (b) \(h\) is greatest when \(\cos x\) is least, \(\cos x = -1\) at \(x = 180^\circ\): \(h = 10 + 8 = 18\) m.
- (c) \(14 = 10 - 8\cos x\) gives \(\cos x = -0.5\).
- \(\cos 60^\circ = 0.5\), so cosine is \(-0.5\) at \(180^\circ - 60^\circ = 120^\circ\) and \(180^\circ + 60^\circ = 240^\circ\).
6 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.
For \(0^\circ \le x \le 360^\circ\), at which values of \(x\) does the graph of \(y = \cos x\) cross the \(x\)-axis?
- A\(90^\circ\) and \(270^\circ\)
- B\(0^\circ\), \(180^\circ\) and \(360^\circ\)
- C\(180^\circ\) only
- D\(45^\circ\) and \(225^\circ\)
The graph shows \(y = \sin x\) for \(0^\circ \le x \le 360^\circ\) and the line \(y = 0.574\). They meet at P, where \(x = 35^\circ\), and at Q. Find the \(x\)-coordinate of Q.
- A\(145^\circ\)
- B\(215^\circ\)
- C\(325^\circ\)
- D\(55^\circ\)
Given that \(\cos 64^\circ = 0.438\), find all the values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\cos x = -0.438\).
- A\(116^\circ\) and \(244^\circ\)
- B\(64^\circ\) and \(296^\circ\)
- C\(116^\circ\) and \(296^\circ\)
- D\(154^\circ\) and \(206^\circ\)
Given that \(\tan 58^\circ = 1.60\), find all the values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\tan x = -1.60\).
- A\(122^\circ\) and \(302^\circ\)
- B\(238^\circ\) and \(302^\circ\)
- C\(122^\circ\) and \(238^\circ\)
- D\(-58^\circ\) and \(122^\circ\)
How many solutions does the equation \(\cos x = 1\) have for \(0^\circ \le x \le 360^\circ\)?
- A2
- B1
- C0
- D4
A seat on a big wheel is \(h\) metres above the ground after the wheel has turned through \(x^\circ\), where \(h = 10 - 8\cos x\). Find the greatest height of the seat and the value of \(x\), with \(0 \le x \le 360\), at which it occurs.
- A18 m when \(x = 180\)
- B18 m when \(x = 0\)
- C10 m when \(x = 90\)
- D2 m when \(x = 180\)
Quiz key
\(\cos x = 0\) at \(x = 90^\circ\) and \(x = 270^\circ\): the curve starts at \((0, 1)\), falls through the axis at \(90^\circ\) to its minimum at \((180^\circ, -1)\), and rises through the axis at \(270^\circ\). The tempting \(0^\circ\), \(180^\circ\), \(360^\circ\) are the zeros of \(\sin x\). \(180^\circ\) is where cosine is \(-1\), and \(45^\circ\), \(225^\circ\) are where \(\sin x = \cos x\).
By the symmetry of the sine curve, \(\sin(180^\circ - x) = \sin x\), so Q is at \(180^\circ - 35^\circ = 145^\circ\). The tempting \(215^\circ\) uses the tangent pattern \(180^\circ + x\), but there \(\sin x\) is negative. \(325^\circ\) uses the cosine pattern \(360^\circ - x\), also a negative sine, and \(55^\circ\) is \(90^\circ - 35^\circ\).
\(\cos x\) is negative in the middle section of the curve, which is symmetrical about \(180^\circ\). The solutions are \(180^\circ - 64^\circ = 116^\circ\) and \(180^\circ + 64^\circ = 244^\circ\). The tempting \(64^\circ\) and \(296^\circ\) solve \(\cos x = +0.438\). \(116^\circ\) and \(296^\circ\) mix one correct value with a positive one, and \(154^\circ\), \(206^\circ\) use \(90^\circ + 64^\circ\) and \(270^\circ - 64^\circ\).
One solution is \(-58^\circ\), outside the range. Adding \(180^\circ\) repeatedly: \(-58^\circ + 180^\circ = 122^\circ\) and \(122^\circ + 180^\circ = 302^\circ\). The tempting \(238^\circ = 180^\circ + 58^\circ\) has \(\tan 238^\circ = +1.60\). \(-58^\circ\) solves the equation but is not between \(0^\circ\) and \(360^\circ\).
The line \(y = 1\) touches the cosine curve at its maximum points, which in this range are \(x = 0^\circ\) and \(x = 360^\circ\), both included by the \(\le\) signs. So there are 2 solutions. The tempting answer 1 forgets the end point \(360^\circ\). 0 confuses this with an impossible equation such as \(\cos x = 1.2\), and 4 assumes every horizontal line cuts the curve as often as possible.
\(h\) is greatest when \(\cos x\) takes its least value, \(-1\), which happens at \(x = 180\). Then \(h = 10 - 8(-1) = 18\) m. The tempting "18 m when \(x = 0\)" has the right height but the wrong angle: at \(x = 0\), \(h = 10 - 8 = 2\) m, the lowest point. 10 m is the height of the centre (at \(x = 90\)), and 2 m is the least height.
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.