GCSE Maths · Higher · Chapter 22: Triangles · Lesson 2

Trigonometric ratios beyond 90°

Grades 7–9About 50 minutesCalculator allowed39 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Write down the exact value of $sin 3 0^{\circ}$ .
[1 mark]
Answer
A2
Last lesson
A triangle has sides 8 cm and 5 cm with $3 0^{\circ}$ between them. Find its area. ( $sin 3 0^{\circ} = 0.5$ )
[1 mark]
Answer
A3
From chapter 19
14 students play chess; 5 of them also play golf. Pick a chess player. Find P(plays golf).
[1 mark]
Answer
A4
From chapter 21
$y \propto \frac{1}{x}$ , and $y = 18$ when $x = 4$ . Find $y$ when $x = 8$ .
[1 mark]
Answer

Grades 7–8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Solve $sin x = 0.46$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.

Hint The sine curve is symmetric about

x = 9 0^{\circ}

B1
Solve $sin x = 0.73$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.
[3 marks]
Answer

Example 2

Solve $cos x = - 0.58$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.

Hint The cosine curve is symmetric about

x = 18 0^{\circ}

B2
Solve $cos x = - 0.43$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.
[3 marks]
Answer

Grades 7–8

C1
Solve $sin x = - 0.62$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.
[3 marks]
Answer
C2
Solve $tan x = 2.7$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.
[2 marks]
Answer

C3
Write down the coordinates of the minimum point of $y = 4 sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[2 marks]
Answer
C4
Solve $5 cos x + 2 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , to 1 d.p.
[3 marks]
Answer
C5
How many solutions does $cos x = 0.8$ have for $0^{\circ} \leq x \leq 72 0^{\circ}$ ?
[1 mark]
Answer
C6
$sin 3 8^{\circ} = 0.616$ to 3 d.p. Without a calculator, write down the value of $sin 14 2^{\circ}$ and the value of $sin 21 8^{\circ}$ .
[2 marks]
Answer

Grades 8–9

D1
Sara solves $tan x = 3.2$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . She writes, " $x = 72. 6^{\circ}$ or $x = 180 - 72.6 = 107. 4^{\circ}$ ."
Explain what Sara has done wrong, and give the correct solutions to 1 d.p.
[3 marks]
Answer

D2
The height, $h$ metres, of a seat on a big wheel $t$ seconds after the ride starts is $h = 13 - 11 cos (6 t)^{\circ}$ , for $0 \leq t \leq 60$ .
(a)
Work out the height of the seat when $t = 10$ .
[1 mark]
Answer
(b)
Write down the greatest height of the seat.
[1 mark]
Answer
(c)
Find the two times when the seat is $20$ m high. Give your answers to 1 d.p.
[4 marks]
Answer
Total for D2: 6 marks
D3
The line $y = k$ meets the curve $y = sin x$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ , at exactly two points. One of them has $x = 2 3^{\circ}$ . Find the $x$ -coordinate of the other point, and the value of $k$ to 3 d.p.
[3 marks]
Answer

Grade 9

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find both solutions of a trig equation between $0^{\circ}$ and $36 0^{\circ}$ .
I can use the symmetry of the sine, cosine and tangent graphs.
I can use a trig model to answer a question in context.

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