Sine rule and cosine ruleIB MYP Maths Extended: Subtopic test

1. 1
   In triangle $ABC$, $B\hat AC=40^\circ$, $A\hat BC=65^\circ$ and $BC=12$ cm.

   (a)

   Find the length of $AC$.

   [1 mark]

   • A$8.51$ cm
   • B$18.0$ cm
   • C$11.3$ cm
   • D$16.9$ cm

   (b)

   Find the length of $AB$.

   [1 mark]

   • A$12.8$ cm
   • B$18.0$ cm
   • C$7.99$ cm
   • D$16.9$ cm

   (c)

   Explain why the cosine rule is not the best first step to find $AC$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In triangle $PQR$, $PQ=6$ cm, $PR=10$ cm and $Q\hat PR=120^\circ$.

   (a)

   Find the length of $QR$.

   [1 mark]

   • A$\sqrt{76}$ cm
   • B$\sqrt{136}$ cm
   • C$14$ cm
   • D$196$ cm

   (b)

   Find the size of angle $P\hat QR$.

   [1 mark]

   • A$38.2^\circ$
   • B$21.8^\circ$
   • C$51.8^\circ$
   • D$141.8^\circ$

   (c)

   Find the exact value of $\cos P\hat QR$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two points $A$ and $B$ are on a straight shoreline, 800 m apart. A boat is at the point $C$ out at sea, with $C\hat AB=52^\circ$ and $C\hat BA=71^\circ$.

   (a)

   Find the distance $AC$ from the boat to the point $A$.

   [3 marks]

   (b)

   A second boat is at the point $D$, where $AD=500$ m and $C\hat AD=34^\circ$. Find the distance $CD$ between the two boats, to the nearest metre.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A robotic arm has two straight sections, of lengths 7 cm and 9 cm, joined at an elbow. The angle between the two sections is $\theta$, and the distance between the two free ends of the arm is $c$ cm.

   (a)

   (i) Find $c^2$ when $\theta=60^\circ$, $90^\circ$ and $120^\circ$.
   (ii) Describe how $c^2$ changes as $\theta$ increases, and explain why $c^2=130$ when $\theta=90^\circ$.

   [6 marks]

   (b)

   (i) A sensor needs the free ends to be 12.5 cm apart. Find $\theta$, to 1 decimal place.
   (ii) Another sensor needs the free ends to be 17 cm apart. Justify whether the arm can do this.

   [6 marks]

   Total for question 4: 12 marks

End of questions