The idea, step by step
Translate the measurements into a model
A baseline and two sightings can form a non-right triangle. Label the stations, measured angles and required side, then mark opposite pairs.
Directions measured at different stations need care: reverse a baseline bearing when necessary. A difference between arbitrary bearing numbers need not be an interior angle.
Select and explain a sequence of methods
Use a sine rule with a complete opposite pair; use cosine rule for SAS or SSS. Some problems need an angle sum, then a side, then an area or perpendicular distance.
Say whether the result is horizontal, sloping or direct distance. Note assumptions such as level ground, straight routes and negligible instrument height; match the final units to the question.
A common trap
Assuming a navigation triangle is right-angled because the sketch looks that way.
Instead: Use a right-triangle method only when a right angle is stated or justified.
See the thinking
Worked example 1
Survey stations A and B are 30 m apart. A tree T lies on one side of AB with angle TAB=55° and TBA=70°. Find AT.
- Angle T=180°−55°−70°=55°.
- AB is opposite T and AT is opposite B.
- AT=30sin70°/sin55°.
Worked example 2
Paths from a gate follow bearings 040° and 115°. P is 18 m along the first and Q is 25 m along the second. Find PQ.
- The included angle at the gate is 115°−40°=75°.
- Use SAS cosine rule.
- PQ²=18²+25²−2×18×25cos75°.
Check your understanding
Two routes from the same point have bearings 025° and 100°. Find their smaller included angle.
Show answer
75°
100°−25°=75° from the same north line.
Your turn
6 questions · 17 marks · Work at your own pace.
Read the questions online
- A 40 m baseline AB has a point P to one side. Angle PAB=35° and PBA=65°. Find AP.3 marks
- Routes from a depot have bearings 020° and 110°. Points 12 km and 16 km along them are joined. Find the joining distance.2 marks
- A triangular plot has sides 30 m and 42 m enclosing 55°. Find its area to one decimal place.2 marks
- Two ground-level observers and the tower base lie on the same straight level line. Both observers are on one side of the tower, 20 m apart. The nearer elevation is 60° and the farther 30°. Find the tower height.4 marks
- P is 20 km from O on bearing 350°. Q is 15 km from O on bearing 030°. Find the smaller angle POQ and PQ.3 marks
- A survey assumes level ground, but a measured baseline slopes up a hill. Explain why its sloping length cannot automatically replace a horizontal baseline.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A 40 m baseline AB has a point P to one side. Angle PAB=35° and PBA=65°. Find AP.3 marks
Angle P=80°.
AP=40sin65°/sin80°.
36.8 m - Routes from a depot have bearings 020° and 110°. Points 12 km and 16 km along them are joined. Find the joining distance.2 marks
The included angle is 90°.
d=√(12²+16²)=20 km.
20 km - A triangular plot has sides 30 m and 42 m enclosing 55°. Find its area to one decimal place.2 marks
K=½×30×42sin55°.
K=630sin55° square metres.
516.1 m² - Two ground-level observers and the tower base lie on the same straight level line. Both observers are on one side of the tower, 20 m apart. The nearer elevation is 60° and the farther 30°. Find the tower height.4 marks
Let x be the nearer horizontal distance. h=x tan60°=(x+20)tan30°.
Solve to obtain x=10 m.
h=10√3 m.
17.3 m - P is 20 km from O on bearing 350°. Q is 15 km from O on bearing 030°. Find the smaller angle POQ and PQ.3 marks
The angle crosses north: 10°+30°=40°.
PQ²=20²+15²−2×20×15cos40°.
40°; PQ≈12.9 km - A survey assumes level ground, but a measured baseline slopes up a hill. Explain why its sloping length cannot automatically replace a horizontal baseline.3 marks
A horizontal triangle uses horizontal distances.
Using an uncorrected sloping distance changes the geometry.
Its horizontal projection is shorter; the model needs a slope correction or additional height/angle information.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗