A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 10: Trigonometric identities and equations · Lesson 3

Solving trigonometric equations

71 marks

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Quick questions to start.

A1
Last lesson
Simplify $\frac{1 - cos ^{2} 2 x}{sin 2 x cos 2 x}$ .
[2 marks]
Answer
A2
Needed today
Write down the two angles between $0^{\circ}$ and $36 0^{\circ}$ whose cosine is $- \frac{1}{2}$ .
[2 marks]
Answer
A3
Needed today
The angle $θ$ is in the interval $0^{\circ} \leq θ \leq 36 0^{\circ}$ . Write down the interval that $2 θ - 4 0^{\circ}$ is in.
[2 marks]
Answer
A4
From chapter 9
In a triangle, $sin B = 0.42$ and angle $B$ is obtuse. Find angle $B$ to 1 decimal place.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it. Give angles to 1 decimal place.

Example 1

Solve $4 cos θ = - 3$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .

Hint Make

cos θ

the subject. The graph of

y = cos θ

is symmetrical about

18 0^{\circ}

, so the second value is

36 0^{\circ}

minus the first.

B1
Solve $9 sin θ + 4 = 0$ for $- 18 0^{\circ} \leq θ \leq 18 0^{\circ}$ .
[3 marks]
Answer $θ = \dots\dots$

Example 2

Solve $3 sin θ = 7 cos θ$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .

Hint Divide both sides by

3 cos θ

to get an equation in

tan θ

. Tan repeats every

18 0^{\circ}

B2
Solve $5 sin x + 3 cos x = 0$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[3 marks]
Answer $x = \dots\dots$

Example 3

Solve $cos 2 θ = 0.3$ for $0^{\circ} \leq θ < 36 0^{\circ}$ . Give your answers to 1 decimal place.

Hint Here

0^{\circ} \leq 2 θ < 72 0^{\circ}

. Find every value of

2 θ

in that interval first, then halve them.

B3
Solve $tan 3 x = - 1.8$ for $0^{\circ} \leq x \leq 18 0^{\circ}$ .
[4 marks]
Answer $x = \dots\dots$

Example 4

Solve $sin (θ + 3 5^{\circ}) = - 0.6$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .

Hint Let

u = θ + 3 5^{\circ}

, so

3 5^{\circ} \leq u < 39 5^{\circ}

. Solve for

u

, then subtract

3 5^{\circ}

B4
Solve $2 tan (θ - 2 5^{\circ}) = 5$ for $- 18 0^{\circ} \leq θ \leq 18 0^{\circ}$ .
[4 marks]
Answer $θ = \dots\dots$

Practice

A-Level

C1
Solve $8 sin^{2} θ = 5$ for $- 18 0^{\circ} \leq θ \leq 18 0^{\circ}$ .
[4 marks]
Answer $θ = \dots\dots$

C2
Find the exact solutions of $2 sin (2 θ + 3 0^{\circ}) = 3$ for $0^{\circ} \leq θ \leq 18 0^{\circ}$ .
[4 marks]
Answer $θ = \dots\dots$
C3
Solve $sin^{2} x = 6 cos^{2} x$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[4 marks]
Answer $x = \dots\dots$
C4
Without solving them, state how many solutions each equation has for $0^{\circ} \leq x < 36 0^{\circ}$ . (a) $sin 4 x = 0.9$ (b) $tan \frac{x}{2} = 3$ (c) $2 cos 3 x = 3$
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Exam-style questions

Edexcel exam style

D1
Kai is asked to solve $2 cos 3 θ = 2$ for $0^{\circ} \leq θ \leq 18 0^{\circ}$ . Kai writes: " $cos 3 θ = \frac{2}{2}$ , so $3 θ = 4 5^{\circ}$ and $θ = 1 5^{\circ}$ . The other solution is $36 0^{\circ} - 1 5^{\circ} = 34 5^{\circ}$ , which is too big. So $θ = 1 5^{\circ}$ is the only solution."
Kai has not found every solution.
(a)
Describe two mistakes in Kai's working.
[2 marks]
(b)
Find all the solutions. Give them exactly.
[4 marks]
Answer $θ = \dots\dots$
Total for D1: 6 marks

D2
The angle $x$ is in the interval $0^{\circ} \leq x < 36 0^{\circ}$ .
(a)
Show that the equation $5 sin (2 x - 2 0^{\circ}) = 2 cos (2 x - 2 0^{\circ})$ can be written as $tan (2 x - 2 0^{\circ}) = 0.4$ .
[2 marks]
(b)
Hence solve $5 sin (2 x - 2 0^{\circ}) = 2 cos (2 x - 2 0^{\circ})$ for $0^{\circ} \leq x < 36 0^{\circ}$ , giving your answers to 1 decimal place.
[4 marks]
Answer $x = \dots\dots$
Total for D2: 6 marks
D3
A seat on a big wheel moves round once every $9$ minutes. Its height above the ground, $h$ metres, $t$ minutes after it leaves the bottom, is modelled by $h = 21 - 18 cos (40 t)^{\circ}$ , for $0 \leq t \leq 9$ .
(a)
Write down the greatest height of the seat, and find the value of $t$ when it is first reached.
[2 marks]
Answer $\dots\dots$ m $t = \dots\dots$
(b)
Find the values of $t$ when the seat is $28$ m above the ground. Give your answers to 2 decimal places.
[4 marks]
Answer $t = \dots\dots$
(c)
Hence find, in minutes, how long the seat is more than $28$ m above the ground in each turn. Give your answer to 2 decimal places.
[2 marks]
Answer $\dots\dots$ minutes
Total for D3: 8 marks
D4
The curve $C$ has equation $y = cos (x + 3 0^{\circ})$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
(a)
Sketch $C$ . Give the coordinates of the points where $C$ meets the coordinate axes, and of its maximum and minimum points.
[4 marks]
(b)
Hence state the number of solutions of $cos (x + 3 0^{\circ}) = - 0.35$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[1 mark]
Answer
(c)
Solve $cos (x + 3 0^{\circ}) = - 0.35$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , giving your answers to 1 decimal place.
[3 marks]
Answer $x = \dots\dots$
Total for D4: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Find every solution of $cos 5 θ = cos θ$ in the interval $0^{\circ} \leq θ \leq 18 0^{\circ}$ . Give your answers exactly.
[6 marks]
Hint 1 · What to try
Two angles have the same cosine when they are the same angle, or minus each other, give or take whole turns of $36 0^{\circ}$ .
Hint 2 · The first line
So $5 θ = θ + 36 0^{\circ} n$ or $5 θ = - θ + 36 0^{\circ} n$ , where $n$ is a whole number.
Hint 3 · The full method
The first gives $θ = 9 0^{\circ} n$ : $0^{\circ}$ , $9 0^{\circ}$ , $18 0^{\circ}$ . The second gives $θ = 6 0^{\circ} n$ : $0^{\circ}$ , $6 0^{\circ}$ , $12 0^{\circ}$ , $18 0^{\circ}$ . Together there are five different solutions.
Answer $θ = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve sin, cos or tan of an angle equals a number in any given interval.
I can turn a sin = b cos into an equation in tan.
I can change the interval first for a multiple or a shifted angle.
I can say how many solutions an equation has, and explain why.

Answers and mark scheme

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