A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 9: Trigonometric ratios · Lesson 1

The cosine rule and the sine rule

70 marks

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Date

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Quick questions to start.

A1
Needed today
A right-angled triangle has hypotenuse $13$ cm. Find the side opposite an angle of $3 5^{\circ}$ , to 3 s.f.
[2 marks]
Answer
A2
Needed today
Solve $x^{2} + 4 x - 17 = 0$ , giving your answers in the form $a \pm b$ .
[2 marks]
Answer $x = \dots\dots$
A3
Last lesson
Estimate $1.0 2^{10}$ using the first three terms of the expansion of $(1 + x)^{10}$ .
[2 marks]
Answer
A4
From chapter 6
Find the centre and the radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 11$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Triangle $P QR$ is shown. Find $QR$ , to 3 s.f.

Hint You know two sides and the angle between them, so use the cosine rule.

cos 11 8^{\circ}

is negative, so the last term adds on.

B1
Triangle $A B C$ is shown. Find $A C$ , to 3 s.f.
[3 marks]
Not drawn accurately
Answer

Example 2

Find the size of the largest angle of this triangle.

Hint The largest angle faces the longest side. Rearrange the cosine rule for

cos θ

B2
Find the size of the smallest angle of this triangle.
[3 marks]
Not drawn accurately
Answer

Example 3

Triangle $A B C$ is shown. Find $A B$ , to 3 s.f.

Hint Find angle

C

first. Pair

A B

with

sin C

and

B C

with

sin A

B3
Triangle $X Y Z$ is shown. Find $Y Z$ , to 3 s.f.
[3 marks]
Not drawn accurately
Answer

Example 4

In triangle $A B C$ , $A B = 9$ cm, $A C = 7$ cm and angle $A B C = 4 2^{\circ}$ . Find the two possible values of angle $A C B$ .

Hint

sin x = sin (18 0^{\circ} - x)

: try the acute and the obtuse angle. Check each fits with the

4 2^{\circ}

B4
In triangle $D E F$ , $D E = 14$ cm, $D F = 11$ cm and angle $D E F = 4 1^{\circ}$ . Find the two possible values of angle $D F E$ , and for each the size of angle $E D F$ .
[4 marks]
Answer

Practice

A-Level

C1
In triangle $D E F$ , $D E = 9.4$ cm, $E F = 12.1$ cm and angle $E D F = 7 1^{\circ}$ . Find angle $D F E$ , and explain why it has only one possible value.
[4 marks]
Not drawn accurately
Answerangle $D F E = \dots\dots$

C2
Triangle $A B C$ is shown, with lengths in centimetres. Find the value of $x$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$
C3
Triangle $A B C$ has $A B = 12$ cm and angle $A B C = 3 5^{\circ}$ . Find the range of values of the length $A C$ for which two different triangles $A B C$ can be drawn.
[4 marks]
Answer $\dots\dots < A C < \dots\dots$
C4
In triangle $A B C$ the sides $a$ , $b$ and $c$ satisfy $a^{2} = b^{2} + c^{2} + b c$ . Show that angle $A = 12 0^{\circ}$ .
[3 marks]

Exam-style questions

Edexcel exam style

D1
A boat leaves harbour $H$ and sails $18$ km on a bearing of $05 2^{\circ}$ to a buoy $A$ . It then sails $25$ km on a bearing of $14 7^{\circ}$ to a buoy $B$ .
Not drawn accurately
(a)
Show that angle $H A B = 8 5^{\circ}$ .
[2 marks]
(b)
Hence find the distance $H B$ , to 3 s.f.
[3 marks]
Answer
(c)
Find the bearing of $H$ from $B$ , to the nearest degree.
[3 marks]
Answer $\dots \dots^{\circ}$
Total for D1: 8 marks

D2
In triangle $A B C$ , $A B = (x + 1)$ cm, $B C = (7 - x)$ cm and angle $A B C = 12 0^{\circ}$ , where $- 1 < x < 7$ .
Not drawn accurately
(a)
Show that $A C^{2} = x^{2} - 6 x + 57$ .
[3 marks]
(b)
Hence find the exact minimum length of $A C$ , and the value of $x$ at which it happens.
[3 marks]
Answer $A C = \dots\dots$ $x = \dots\dots$
Total for D2: 6 marks
D3
In triangle $P QR$ , $P Q = 11$ cm, $QR = 8$ cm and angle $QP R = 4 0^{\circ}$ . Kai writes: $sin R = \frac{11 sin 4 0 ^{\circ}}{8} = 0.884$ , so angle $P R Q = 62. 1^{\circ}$ .
Kai says this is the only possible value. Kai is wrong.
(a)
Explain the mistake Kai has made.
[2 marks]
(b)
Find the size of angle $P QR$ in each of the two triangles.
[2 marks]
Answer $\dots\dots$ or $\dots\dots$
(c)
Hence find the two possible lengths of $P R$ , to 3 s.f.
[4 marks]
Answer $\dots\dots$ cm or $\dots\dots$ cm
Total for D3: 8 marks
D4
The diagram shows a field $A B C D$ . $A B = 85$ m, $B C = 62$ m, $C D = 74$ m, $D A = 96$ m and angle $B A D = 7 8^{\circ}$ .
Not drawn accurately
(a)
Find the length of the diagonal $B D$ , to 3 s.f.
[3 marks]
Answer
(b)
Hence find the size of angle $B C D$ .
[3 marks]
Answer
Total for D4: 6 marks

Extension

Stretch

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

E1
The sides of a triangle are three consecutive whole numbers of centimetres, and its largest angle is twice its smallest angle. Find the lengths of the three sides.
[6 marks]
Hint 1 · What to try
Bisect the largest angle. Look for an isosceles triangle and a pair of similar triangles.
Hint 2 · The first line
Smallest angle $θ$ at $A$ (side $a = n - 1$ ), angle $2 θ$ at $B$ (side $b = n + 1$ ), $c = n$ . The bisector of angle $B$ meets $A C$ at $D$ : $A D = B D$ , and triangle $B D C$ is similar to triangle $A B C$ .
Hint 3 · The full method
Similar triangles: $D C = \frac{a ^{2}}{b}$ , $A D = B D = \frac{a c}{b}$ . $A D + D C = b$ gives $b^{2} = a (a + c)$ , so $(n + 1)^{2} = (n - 1) (2 n - 1)$ , $n = 5$ : sides $4$ , $5$ , $6$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the cosine rule to find a side or an angle.
I can use the sine rule to find a side or an angle.
I can find both angles when the sine rule gives two.
I can solve problems with bearings and algebraic sides.

Answers and mark scheme

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