The shape of the lesson
By the end of the lesson, learners can
- Every solution in the range: In the lab, drag the horizontal line \(y = k\) up and down across the graph and watch the solutions move in pairs.
- Negative values and tangent: For a negative value of \(\sin x\) or \(\tan x\), the calculator gives a negative angle, which is outside the range.
- No solutions, and exact solutions: Sine and cosine can never be bigger than 1 or less than \(-1\), so some equations have no solutions at all.
- Rearrange to \(\sin x = k\), \(\cos x = k\) or \(\tan x = k\).
- Check that \(-1 \le k \le 1\) for sine and cosine; otherwise no solutions.
- First angle from \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\).
- Other angles: \(180^\circ - \alpha\) (sin), \(360^\circ - \alpha\) (cos), \(\alpha + 180^\circ\) (tan).
- Range: keep only \(0^\circ \le x \le 360^\circ\); give answers to 1 d.p.
One picture every idea returns to
Rearrange to \(\sin x = k\), \(\cos x = k\) or \(\tan x = k\). Use the inverse key for the first angle, then the graph's symmetry for the rest: \(\sin\): \(180^\circ - \alpha\); \(\cos\): \(360^\circ - \alpha\); \(\tan\): \(\alpha + 180^\circ\). Keep only answers in \(0^\circ \le x \le 360^\circ\).
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, One calculator answer, two solutions.. 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: Every solution in the range | In the lab, drag the horizontal line \(y = k\) up and down across the graph and watch the solutions move in pairs. Project tab 3 of the lab (Trigonometric graphs lab, solving equations) and let a student drive it while the class predicts. Model the worked example Solving a sine equation on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Negative values and tangent | For a negative value of \(\sin x\) or \(\tan x\), the calculator gives a negative angle, which is outside the range. Model the worked example Tangent with a negative value on the board, one line at a time. Your turn: Solve \(4\cos x - 1 = 0\) for \(0^\circ \le x \le 360^\circ\). | Pairs |
| 21 to 30 min | Teach and check: No solutions, and exact solutions | Sine and cosine can never be bigger than 1 or less than \(-1\), so some equations have no solutions at all. Model the worked example An impossible equation on the board, one line at a time. | Round Table |
| 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 No solutions, and exact solutions, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| Stopping after the calculator answer. 'Solve for \(0^\circ \le x \le 360^\circ\)' almost always has two answers, and the second one is a mark on its own. | Return to Every solution in the range: ask a pair to state the correct version and show it on the lab before moving on. |
| Giving \(-63.4^\circ\) as an answer. | It solves the equation but is outside \(0^\circ \le x \le 360^\circ\), so it earns nothing. Use it only as a stepping stone. |
| Using the pattern for the wrong function. | For \(\cos x = 0.25\), the partner of \(75.5^\circ\) is \(360^\circ - 75.5^\circ\), not \(180^\circ - 75.5^\circ\). A quick sketch of the right graph prevents this. |
Solving trigonometric equations: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Without a calculator, solve \(\sin x = \dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).
Solve \(\cos x = 0.35\) for \(0^\circ \le x \le 360^\circ\).
Solve \(3\tan x - 4 = 0\) for \(0^\circ \le x \le 360^\circ\).
Solve \(4\sin x + 3 = 0\) for \(0^\circ \le x \le 360^\circ\).
(a) Explain why the equation \(2\cos x = 3\) has no solutions.
(b) Without a calculator, solve \(2\cos x = -\sqrt{3}\) for \(0^\circ \le x \le 360^\circ\).
The depth of water, \(d\) metres, at a harbour entrance is modelled by \(d = 6 + 3\sin x\), where \(x\) is measured in degrees and \(0 \le x \le 360\) covers one tidal cycle.
(a) Find the greatest depth.
(b) A ship needs a depth of at least 7.5 m. Find the values of \(x\) between which the ship can safely enter.
Answer key and marking notes
- \(\sin 30^\circ = \dfrac{1}{2}\), so \(x = 30^\circ\).
- Second solution: \(180^\circ - 30^\circ = 150^\circ\).
- \(x = \cos^{-1}(0.35) = 69.5^\circ\)
- \(360^\circ - 69.5^\circ = 290.5^\circ\)
- \(\tan x = \dfrac{4}{3}\)
- \(x = \tan^{-1}\left(\dfrac{4}{3}\right) = 53.1^\circ\)
- \(53.1^\circ + 180^\circ = 233.1^\circ\)
- \(\sin x = -0.75\); the calculator gives \(-48.6^\circ\), outside the range.
- Acute angle with \(\sin = 0.75\) is \(48.6^\circ\); sine is negative from \(180^\circ\) to \(360^\circ\).
- \(x = 180^\circ + 48.6^\circ = 228.6^\circ\) or \(x = 360^\circ - 48.6^\circ = 311.4^\circ\)
- (a) \(\cos x = 1.5\), but \(\cos x\) can never be greater than 1.
- (b) \(\cos x = -\dfrac{\sqrt{3}}{2}\). Since \(\cos 30^\circ = \dfrac{\sqrt{3}}{2}\) and cosine is negative from \(90^\circ\) to \(270^\circ\):
- \(x = 180^\circ - 30^\circ = 150^\circ\) or \(x = 180^\circ + 30^\circ = 210^\circ\)
- (a) Greatest when \(\sin x = 1\): \(d = 6 + 3 = 9\) m.
- (b) Solve \(6 + 3\sin x = 7.5\): \(\sin x = 0.5\).
- \(x = 30^\circ\) or \(x = 180^\circ - 30^\circ = 150^\circ\).
- Between these the sine graph is above 0.5, so the ship can enter for \(30 \le x \le 150\).
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.
Solve \(\sin x = 0.35\) for \(0^\circ \le x \le 360^\circ\).
- A\(20.5^\circ\) and \(159.5^\circ\)
- B\(20.5^\circ\) only
- C\(20.5^\circ\) and \(339.5^\circ\)
- D\(20.5^\circ\) and \(200.5^\circ\)
Solve \(3\cos x - 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
- A\(70.5^\circ\) and \(289.5^\circ\)
- B\(109.5^\circ\) and \(250.5^\circ\)
- C\(70.5^\circ\) and \(109.5^\circ\)
- D\(48.2^\circ\) and \(311.8^\circ\)
Solve \(4\tan x + 7 = 0\) for \(0^\circ \le x \le 360^\circ\).
- A\(119.7^\circ\) and \(299.7^\circ\)
- B\(60.3^\circ\) and \(240.3^\circ\)
- C\(-60.3^\circ\) and \(119.7^\circ\)
- D\(119.7^\circ\) and \(240.3^\circ\)
Solve \(5\sin x + 2 = 0\) for \(0^\circ \le x \le 360^\circ\).
- A\(203.6^\circ\) and \(336.4^\circ\)
- B\(23.6^\circ\) and \(156.4^\circ\)
- C\(-23.6^\circ\) and \(203.6^\circ\)
- D\(156.4^\circ\) and \(203.6^\circ\)
Which of these equations has NO solutions for \(0^\circ \le x \le 360^\circ\)?
- A\(4\cos x + 7 = 2\)
- B\(2\sin x + 1 = 0\)
- C\(\tan x = 8\)
- D\(3\cos x = -2\)
Without a calculator, solve \(2\cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
- A\(120^\circ\) and \(240^\circ\)
- B\(60^\circ\) and \(300^\circ\)
- C\(60^\circ\) and \(120^\circ\)
- D\(210^\circ\) and \(330^\circ\)
Quiz key
\(x = \sin^{-1} 0.35 = 20.49\ldots = 20.5^\circ\). The sine curve is symmetrical about \(90^\circ\), so the second solution is \(180^\circ - 20.5^\circ = 159.5^\circ\). The tempting "\(20.5^\circ\) only" stops at the calculator answer and loses a mark. \(339.5^\circ\) uses the cosine pattern and \(200.5^\circ\) the tangent pattern; sine is negative at both.
\(3\cos x = 1\), so \(\cos x = \dfrac{1}{3}\) and \(x = \cos^{-1}\dfrac{1}{3} = 70.5^\circ\). For cosine the partner is \(360^\circ - 70.5^\circ = 289.5^\circ\). The tempting \(70.5^\circ\) and \(109.5^\circ\) use the SINE pattern \(180^\circ - \alpha\). \(109.5^\circ\), \(250.5^\circ\) solve \(\cos x = -\dfrac{1}{3}\) (a sign slip), and \(48.2^\circ\), \(311.8^\circ\) solve \(\cos x = \dfrac{2}{3}\) from adding the 1 to the wrong side.
\(\tan x = -\dfrac{7}{4} = -1.75\), so the calculator gives \(-60.3^\circ\). Tangent repeats every \(180^\circ\): \(-60.3^\circ + 180^\circ = 119.7^\circ\) and \(119.7^\circ + 180^\circ = 299.7^\circ\). The tempting \(60.3^\circ\), \(240.3^\circ\) solve \(\tan x = +1.75\). \(-60.3^\circ\) is outside the range, and \(240.3^\circ\) has a positive tangent.
\(\sin x = -0.4\); the calculator gives \(-23.6^\circ\). The solutions in range are \(180^\circ + 23.6^\circ = 203.6^\circ\) and \(360^\circ - 23.6^\circ = 336.4^\circ\). The tempting \(23.6^\circ\), \(156.4^\circ\) solve \(\sin x = +0.4\). \(-23.6^\circ\) is outside the range, and \(156.4^\circ\) has a positive sine.
\(4\cos x + 7 = 2\) gives \(\cos x = -\dfrac{5}{4} = -1.25\), which is less than \(-1\), so there are no solutions. \(2\sin x + 1 = 0\) gives \(\sin x = -0.5\) (two solutions). The tempting \(\tan x = 8\) looks too big, but tangent takes every value, so it has two solutions. \(3\cos x = -2\) gives \(\cos x = -0.667\), which is allowed.
\(\cos x = -\dfrac{1}{2}\). Since \(\cos 60^\circ = \dfrac{1}{2}\), the solutions where cosine is negative are \(180^\circ - 60^\circ = 120^\circ\) and \(180^\circ + 60^\circ = 240^\circ\). The tempting \(60^\circ\), \(300^\circ\) solve \(\cos x = +\dfrac{1}{2}\). \(60^\circ\), \(120^\circ\) use the sine pattern, and \(210^\circ\), \(330^\circ\) solve \(\sin x = -\dfrac{1}{2}\).
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.