TrigonometryIB MYP Maths Standard: Topic test

1. 1
   A wheelchair ramp at a library in Wellington is a straight slope $5$ m long. It rises $0.6$ m vertically from the ground to the door. Let $\theta$ be the angle between the ramp and the horizontal ground.

   (a)

   Which equation correctly links $\theta$ to the lengths in the ramp?

   [1 mark]

   • A$\cos\theta=\frac{0.6}{5}$
   • B$\sin\theta=\frac{0.6}{5}$
   • C$\tan\theta=\frac{0.6}{5}$
   • D$\sin\theta=\frac{5}{0.6}$

   (b)

   Calculate $\theta$ to the nearest degree.

   [1 mark]

   • A$83^\circ$
   • B$12^\circ$
   • C$0.1^\circ$
   • D$7^\circ$

   (c)

   Calculate the horizontal distance from the foot of the ramp to the point directly below the door. Give your answer to $3$ significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A kite is flying on a taut straight string $40$ m long. The string makes an angle of $35^\circ$ with level ground. Ignore the height of the person holding the string.

   (a)

   Find the height of the kite above the ground.

   [1 mark]

   • A$22.9$ m
   • B$32.8$ m
   • C$28.0$ m
   • D$69.7$ m

   (b)

   What is the angle of depression of the person holding the string, as seen from the kite?

   [1 mark]

   • A$55^\circ$
   • B$90^\circ$
   • C$35^\circ$
   • D$145^\circ$

   (c)

   More string is let out until it is $60$ m long. It makes the same angle of $35^\circ$ with the ground. Calculate the new height of the kite, to $3$ significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle $XYZ$ the angle at $Y$ is $90^\circ$, $XY=9$ cm and $YZ=12$ cm.

   (a)

   Calculate angle $YXZ$. Give your answer to $1$ decimal place.

   [3 marks]

   (b)

   The point $W$ lies on $XZ$ so that $YW$ is perpendicular to $XZ$. Calculate the length of $YW$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone hovers $30$ m above level ground. Two people, Ines and Jomo, stand on the ground in a straight line below the drone, on the same side of the point directly under it. From the drone, the angle of depression of Ines is $52^\circ$ and the angle of depression of Jomo is $28^\circ$.

   (a)

   (i) Explain why the angle of elevation of the drone from Ines is $52^\circ$. (ii) Calculate the horizontal distance from the point directly below the drone to Ines, and to Jomo, each to $3$ significant figures. (iii) Hence find the distance between Ines and Jomo.

   [6 marks]

   (b)

   The drone then flies in a straight line directly to Jomo at a constant speed of $4$ m/s. (i) Calculate the straight-line distance from the drone to Jomo. (ii) Calculate the time the flight takes. (iii) The battery warning shows $20$ seconds of flying time left. Decide whether the drone can reach Jomo before the battery runs out, and state one assumption in your calculation.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A helicopter flies from a heliport $H$ in Dubai on a straight path to an oil platform $P$. The bearing of $P$ from $H$ is $250^\circ$ and the distance $HP$ is $180$ km.

   (a)

   What is the bearing of $H$ from $P$?

   [1 mark]

   • A$110^\circ$
   • B$160^\circ$
   • C$340^\circ$
   • D$070^\circ$

   (b)

   A scale drawing uses $1$ cm to represent $20$ km. How long is the line $HP$ on the drawing?

   [1 mark]

   • A$9$ cm
   • B$90$ cm
   • C$0.9$ cm
   • D$3\,600$ cm

   (c)

   A second platform $Q$ is on a bearing of $025^\circ$ from $H$. Find the size of angle $PHQ$, the angle that is less than $180^\circ$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A drone leaves a launch point $L$ in Rotterdam and flies $6$ km due south to a point $M$, then $8$ km due west to a point $N$.

   (a)

   How far is $N$ from $L$ in a straight line?

   [1 mark]

   • A$14$ km
   • B$2$ km
   • C$10$ km
   • D$100$ km

   (b)

   What is the bearing of $N$ from $L$, to the nearest degree?

   [1 mark]

   • A$307^\circ$
   • B$233^\circ$
   • C$127^\circ$
   • D$037^\circ$

   (c)

   Find the bearing of $L$ from $N$, to the nearest degree.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A search plane leaves an airfield $A$ and flies $80$ km on a bearing of $060^\circ$ to a point $B$. It then flies $30$ km due east to a point $C$.

   (a)

   Calculate how far east and how far north $B$ is from $A$. Give each answer to $3$ significant figures.

   [3 marks]

   (b)

   Calculate the bearing of $C$ from $A$, to the nearest degree.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A ferry leaves port $P$, sails $24$ km on a bearing of $110^\circ$ to an island $I$, then sails $18$ km on a bearing of $200^\circ$ to an island $J$.

   (a)

   (i) Show that angle $PIJ$ is $90^\circ$. (ii) Calculate the distance $PJ$. (iii) Calculate the bearing of $J$ from $P$, to the nearest degree.

   [6 marks]

   (b)

   The ferry uses $1.8$ litres of fuel for every kilometre. A direct route from $P$ to $J$ is proposed, but it passes close to a protected coral reef, while the route through $I$ lets passengers visit the island's market. (i) Calculate the fuel used on the route $P\to I\to J$. (ii) Calculate the fuel saved by using the direct route. (iii) Evaluate which route the company should use, referring to your answers and one non-mathematical factor.

   [6 marks]

   Total for question 8: 12 marks

End of questions