Trigonometry in motion
Sailors, pilots, surveyors, engineers and physicists use trigonometry every day to find distances, directions and forces they can’t measure directly.
Bearings
A bearing gives direction as an angle clockwise from north, written with three digits.
| Bearing | Direction |
|---|---|
| 000° | north |
| 090° | east |
| 180° | south |
| 270° | west |
| 045° | northeast |
Navigation example
A ship sails 50 km on a bearing of 040°, then 70 km on a bearing of 120°. How far is it from port?
- The angle between the two legs at the turn: 180° − (120° − 40°) = 100°
- Cosine law: d² = 50² + 70² − 2(50)(70) cos 100°
- d² = 2,500 + 4,900 + 1,215.5 ≈ 8,615.5
- d ≈ 92.8 km
Breaking vectors into components
A vector (like a velocity or a force) can be split into components:
- Horizontal: v cos θ
- Vertical: v sin θ
(θ measured from the horizontal.)
Example: A ball is kicked at 20 m/s at 35° above the ground.
- Horizontal: 20 cos 35° ≈ 16.4 m/s
- Vertical: 20 sin 35° ≈ 11.5 m/s
Resultant vectors
The resultant is the combined effect. Add components, then use Pythagoras and tan⁻¹.
Example: A plane flies north at 300 km/h; the wind blows east at 40 km/h.
- Ground speed: √(300² + 40²) ≈ 302.7 km/h
- Direction: tan⁻¹(40/300) ≈ 7.6° east of north → bearing 008°
Inclined planes
On an inclined plane at angle θ, gravity (mg) splits into:
- Along the slope: mg sin θ (pulls the object down the ramp)
- Into the slope: mg cos θ
Example: A 50 kg box on a 20° ramp. Force along the ramp ≈ 50 × 9.8 × sin 20° ≈ 168 N.
Physics: waves and circular motion
- Alternating current: V(t) = V₀ sin(ωt)
- A point on a rotating wheel: x = r cos θ, y = r sin θ
- Pendulums and springs: sinusoidal motion
Surveying
Example: From two points 100 m apart on a straight road, the angles to a tower are 32° and 47°. Use the sine law in the triangle formed to find the distance to the tower, then trig ratios for its height.
Worked examples
Example 1: Bearing for southwest? 225°.
Example 2: A force of 80 N at 60° to the horizontal. Horizontal component? 80 cos 60° = 40 N.
Example 3: Walk 3 km east and 4 km north. Resultant? 5 km, bearing tan⁻¹(3/4) ≈ 037°.
Common mistakes
- Measuring bearings counterclockwise or from east.
- Using sin for the horizontal component when θ is from the horizontal.
- Forgetting to find direction along with size.
Try it yourself
- Write the bearing for due west.
- Velocity 25 m/s at 40°: horizontal and vertical components?
- A boat heads east at 12 km/h across a current flowing south at 5 km/h. Resultant speed?
- Force along a 30° ramp for a 10 kg mass (g = 9.8)?
Answers: 1) 270° 2) ≈ 19.2 m/s and ≈ 16.1 m/s 3) 13 km/h 4) 49 N
Quick recap
- Bearings are measured clockwise from north.
- Split a vector into components with cos and sin; combine them into a resultant.
- Trig solves navigation, force and inclined plane problems.