Checkpoint 1 of 3 10 MC + 2 Short Answer Covers Lessons 1-7

Checkpoint 1

Right-angled Trigonometry. Assess your understanding of sine, cosine, tangent, finding unknown sides and angles, angles of elevation and depression, and compass bearings.

Instructions: Complete all questions. For multiple choice, select one answer and click Check. For short answer, show all working. Your responses are saved automatically.
Work mode:

Section A - Multiple Choice

2 marks each. Select the best answer.

2 marks In a right-angled triangle, $\sin \theta = \frac{3}{5}$. The hypotenuse is 20 cm. The length of the opposite side is:

2 marks If $\cos \theta = 0.6$, then $\theta$ is approximately:

2 marks A tower casts a shadow 30 m long. The angle of elevation of the sun is $35°$. The height of the tower is closest to:

2 marks The angle of depression from the top of a cliff to a boat is $25°$. If the cliff is 80 m high, the distance from the base of the cliff to the boat is closest to:

2 marks A bearing of $135°$ is equivalent to the compass direction:

2 marks The exact value of $\tan 45°$ is:

2 marks A ship sails $15$ km on a bearing of $070°$. The distance it has travelled east is closest to:

2 marks In a right-angled triangle, the hypotenuse is $17$ cm and one side is $8$ cm. The other side is:

2 marks The back bearing of $210°$ is:

2 marks A ramp makes an angle of $8°$ with the horizontal. If the ramp is $5$ m long, the vertical rise is closest to:

Section B - Short Answer

Show all working. 5 marks each.

Question 11

5 marks Apply / Analyse

From the top of a $45$ m tall lighthouse, the angle of depression to a boat is $18°$.

(a) Draw a diagram showing the angle of depression and the right-angled triangle formed. (1 mark)

(b) Calculate the horizontal distance from the base of the lighthouse to the boat. (2 marks)

(c) A second boat is directly behind the first boat, and the angle of depression to the second boat is $12°$. How far apart are the two boats? (2 marks)

Marking Criteria

  • 1 mark: Correct diagram with angle of depression shown (equal to angle of elevation from boat)
  • 1 mark: Set up trig ratio $\tan 18° = \frac{45}{d}$
  • 1 mark: Correct distance $d = \frac{45}{\tan 18°} \approx 138$ m
  • 1 mark: Distance to second boat $= \frac{45}{\tan 12°} \approx 212$ m
  • 1 mark: Difference $212 - 138 = 74$ m (accept 70-80 m)

Question 12

5 marks Analyse / Evaluate

A hiker walks $4$ km on a bearing of $060°$, then turns and walks $3$ km on a bearing of $150°$.

(a) Sketch the hiker's journey, showing both legs and the angle between them. (1 mark)

(b) Show that the angle between the two paths is $90°$. (1 mark)

(c) Calculate how far the hiker is from their starting point. (2 marks)

(d) On what bearing must the hiker walk to return directly to the start? (1 mark)

Marking Criteria

  • 1 mark: Accurate sketch with North reference and bearings marked
  • 1 mark: Angle between paths $= 150° - 60° = 90°$
  • 1 mark: Apply Pythagoras: $d^2 = 4^2 + 3^2 = 25$
  • 1 mark: Correct distance $d = 5$ km
  • 1 mark: Correct back bearing calculation (accept $240°$ or reasoning about return direction)