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Assessment · Sine & Cosine Rule
20 marks
Answer all questions. Show all your working; you get marks for method. Work on your own.Name: __________ Class: ______ Date: ______
In triangle $ABC$, angle $A = 50^\circ$, angle $B = 72^\circ$ and side $a = 13$ cm. Work out the length of side $b$, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, $b = 11$ cm, $c = 10$ cm and the angle $A = 119^\circ$ between them. Work out the length of side $a$, to 1 decimal place.
Answer: (3 marks)
A triangle has two sides of length 6 cm and 14 cm with an angle of 136$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, angle $A = 55^\circ$, angle $B = 62^\circ$ and side $a = 8$ cm. Work out the length of side $b$, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, $b = 12$ cm, $c = 10$ cm and the angle $A = 94^\circ$ between them. Work out the length of side $a$, to 1 decimal place.
Answer: (3 marks)
A triangle has two sides of length 8 cm and 9 cm with an angle of 102$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, angle $A = 37^\circ$, angle $B = 73^\circ$ and side $a = 9$ cm. Work out the length of side $b$, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, $b = 13$ cm, $c = 6$ cm and the angle $A = 114^\circ$ between them. Work out the length of side $a$, to 1 decimal place.
Answer: (3 marks)
A triangle has two sides of length 6 cm and 11 cm with an angle of 81$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.
Answer: (3 marks)
In triangle $ABC$, angle $A = 67^\circ$, side $a = 9$ cm and side $b = 6$ cm. Side $b$ is shorter than side $a$, so angle $B$ is acute. Work out the size of angle $B$, to 1 decimal place.
Answer: (3 marks)
A triangle $ABC$ has sides $a = 15$ cm, $b = 9$ cm and $c = 8$ cm. Work out the size of angle $A$, to 1 decimal place.
Answer: (3 marks)
Triangle $ABC$ has $b = 12$ cm, $c = 16$ cm and the included angle $A = 106^\circ$. Work out the perimeter of the triangle, to 1 decimal place.
Answer: (4 marks)
A triangle has area 52.6 cm². One side is 13 cm and the angle between that side and an unknown side $b$ is 54$^\circ$. Work out the length of side $b$, to 1 decimal place.
Answer: (3 marks)
A ship sails 10 km from $P$ to $Q$ on a bearing of $050^\circ$. At $Q$ it changes course and sails 10 km to $R$ on a bearing of $148^\circ$. Work out the direct distance from $P$ to $R$, to 1 decimal place.
Answer: (4 marks)
In triangle $ABC$, angle $B = 69^\circ$, angle $C = 60^\circ$ and side $c = AB = 13$ cm. Work out the area of triangle $ABC$, to 1 decimal place.
Answer: (5 marks)
A walker leaves camp ($A$) and walks 13 km to the lake ($B$) on a bearing of $093^\circ$. From the lake the walker turns and walks 13 km to the summit ($C$) on a bearing of $191^\circ$. Work out the bearing of the summit from camp, to the nearest degree.
Answer: (5 marks)
Mark scheme · Sine & Cosine Rule
Total: 20 marks
Award the marks shown for each correct step, then add up the total out of 20. A method mark counts even if the final answer is wrong.
In triangle $ABC$, angle $A = 50^\circ$, angle $B = 72^\circ$ and side $a = 13$ cm. Work out the length of side $b$, to 1 decimal place.[3 marks]
Method
Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$.
A triangle has two sides of length 6 cm and 14 cm with an angle of 136$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.[3 marks]
Method
Area $= \dfrac{1}{2}ab\sin C$.
$= \dfrac{1}{2}(6)(14)\sin 136^\circ$
$= 29.2$ cm².
Answer: $29.2$
Marks
✔1 markArea = ½ab sin C
✔1 markSubstitute the two sides and included angle
✔1 mark29.2 cm²
In triangle $ABC$, angle $A = 55^\circ$, angle $B = 62^\circ$ and side $a = 8$ cm. Work out the length of side $b$, to 1 decimal place.[3 marks]
Method
Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$.
A triangle has two sides of length 8 cm and 9 cm with an angle of 102$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.[3 marks]
Method
Area $= \dfrac{1}{2}ab\sin C$.
$= \dfrac{1}{2}(8)(9)\sin 102^\circ$
$= 35.2$ cm².
Answer: $35.2$
Marks
✔1 markArea = ½ab sin C
✔1 markSubstitute the two sides and included angle
✔1 mark35.2 cm²
In triangle $ABC$, angle $A = 37^\circ$, angle $B = 73^\circ$ and side $a = 9$ cm. Work out the length of side $b$, to 1 decimal place.[3 marks]
Method
Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$.
A triangle has two sides of length 6 cm and 11 cm with an angle of 81$^\circ$ between them. Work out the area of the triangle, to 1 decimal place.[3 marks]
Method
Area $= \dfrac{1}{2}ab\sin C$.
$= \dfrac{1}{2}(6)(11)\sin 81^\circ$
$= 32.6$ cm².
Answer: $32.6$
Marks
✔1 markArea = ½ab sin C
✔1 markSubstitute the two sides and included angle
✔1 mark32.6 cm²
In triangle $ABC$, angle $A = 67^\circ$, side $a = 9$ cm and side $b = 6$ cm. Side $b$ is shorter than side $a$, so angle $B$ is acute. Work out the size of angle $B$, to 1 decimal place.[3 marks]
Method
Sine rule: $\dfrac{\sin B}{b} = \dfrac{\sin A}{a}$.
A triangle $ABC$ has sides $a = 15$ cm, $b = 9$ cm and $c = 8$ cm. Work out the size of angle $A$, to 1 decimal place.[3 marks]
Method
Cosine rule: $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$.
$\cos A = \dfrac{9^2 + 8^2 - 15^2}{2(9)(8)} = -0.5556$.
$A = \cos^{-1}(-0.5556) = 123.7^\circ$.
Answer: $123.7$
Marks
✔1 markRearranged cosine rule for cos A
✔1 markcos A = -0.556
✔1 mark123.7°
Triangle $ABC$ has $b = 12$ cm, $c = 16$ cm and the included angle $A = 106^\circ$. Work out the perimeter of the triangle, to 1 decimal place.[4 marks]
Method
Find $a$ with the cosine rule: $a^2 = 12^2 + 16^2 - 2(12)(16)\cos 106^\circ = 505.84$.
$a = \sqrt{505.84} = 22.49$ cm.
Perimeter $= 22.49 + 12 + 16 = 50.5$ cm.
Answer: $50.5$
Marks
✔1 markCosine rule with correct values
✔1 marka² = 505.8
✔1 marka = 22.5 cm
✔1 mark50.5 cm
A triangle has area 52.6 cm². One side is 13 cm and the angle between that side and an unknown side $b$ is 54$^\circ$. Work out the length of side $b$, to 1 decimal place.[3 marks]
Method
Area $= \dfrac{1}{2}ab\sin C$, so $52.6 = \dfrac{1}{2}(13)\,b\,\sin 54^\circ$.
A ship sails 10 km from $P$ to $Q$ on a bearing of $050^\circ$. At $Q$ it changes course and sails 10 km to $R$ on a bearing of $148^\circ$. Work out the direct distance from $P$ to $R$, to 1 decimal place.[4 marks]
Method
The interior angle at $Q$ is $180^\circ - 98^\circ = 82^\circ$ (the ship turns through $98^\circ$).
In triangle $ABC$, angle $B = 69^\circ$, angle $C = 60^\circ$ and side $c = AB = 13$ cm. Work out the area of triangle $ABC$, to 1 decimal place.[5 marks]
Sine rule for $a = BC$: $a = \dfrac{c \sin A}{\sin C} = \dfrac{13 \times \sin 51^\circ}{\sin 60^\circ} = 11.67$ cm.
Area $= \dfrac{1}{2}\,(AB)(BC)\sin B = \dfrac{1}{2}(13)(11.67)\sin 69^\circ$
$= 70.8$ cm².
Answer: $70.8$
Marks
✔1 markA = 51°
✔1 markSine rule set up for the missing side
✔1 marka = 11.7 cm
✔1 markArea = ½(AB)(BC) sin B
✔1 mark70.8 cm²
A walker leaves camp ($A$) and walks 13 km to the lake ($B$) on a bearing of $093^\circ$. From the lake the walker turns and walks 13 km to the summit ($C$) on a bearing of $191^\circ$. Work out the bearing of the summit from camp, to the nearest degree.[5 marks]
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