Module 2/5 · Weeks 4–6 · 27 h

Trigonometry and angles in flight

DRT 111 Applied Mathematics for Technology

About 80 minDraft, awaiting reviewLast updated 26 September 2026

By the end of this module you will be able to

Convert angles between degrees and radians
Use Pythagoras' theorem and the sine, cosine and tangent ratios to solve right triangles
Find a bearing from coordinate differences with atan2, in the correct quadrant
Calculate the ground footprint and ground sampling distance (GSD) of a drone camera
Resolve wind into headwind and crosswind components

Prerequisites: DRT 111 Module 1 (units and rearranging equations)

Angles are part of every flight

A drone climbs at an angle. A camera looks down with a certain field of view. Wind blows across the flight path at an angle, and pilots give directions as an angle from north. All of these use the same tool: trigonometry, the study of how the angles and side lengths of triangles relate.

By the end of this module you will be able to work out how wide an area a camera at 100 m covers, which is the basis of survey flight planning in DRT 451 and DRT 452.

Degrees and radians

Angles can be measured in two units. Degrees divide a circle into 360 parts. Radians are defined by arc length: one radian is the angle whose arc is exactly as long as the radius. A full circle is therefore $2 π$ radians. The relationship to remember is

18 0^{\circ} = π rad \Rightarrow θ_{rad} = θ_{deg} \times \frac{π}{180}

For example, $3 0^{\circ} = 30 \times π /180 = π /6 \approx 0.524$ rad.

Why do radians matter? Almost all software, including Python and the PX4 and ArduPilot autopilots, works in radians internally. If you pass in degrees without converting, every result will be wrong.

Watch out

Calculators have DEG and RAD modes. Check the mode before pressing sin, cos or tan. $sin 30$ is 0.5 in DEG mode but −0.988 in RAD mode.

The right triangle

A right triangle has one $9 0^{\circ}$ angle. The longest side, opposite the right angle, is the hypotenuse ( $c$ ). Once you choose an angle $θ$ to work with, the other sides are the opposite ( $a$ ) and the adjacent ( $b$ ).

Right triangle with angle theta at the lower left, adjacent side b along the bottom, opposite side a on the right and hypotenuse c as the sloping side — Figure 1. Naming the sides of a right triangle relative to angle θ

Pythagoras’ theorem

c^{2} = a^{2} + b^{2}

Trigonometric ratios

sin θ = \frac{a}{c} cos θ = \frac{b}{c} tan θ = \frac{a}{b}

A popular memory aid is “SOH-CAH-TOA”: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

If you know a ratio and want the angle, use the inverse function. For example, if $tan θ = 0.5$ , then $θ = tan^{- 1} (0.5) \approx 26. 6^{\circ}$ .

Example 1. Climb angle

A fixed-wing drone travels 300 m horizontally while climbing 100 m. What are the climb angle and the actual distance flown?

Climb angle: $θ = tan^{- 1} \frac{100}{300} = tan^{- 1} (0.333) \approx 18. 4^{\circ}$
Distance flown: $c = 30 0^{2} + 10 0^{2} = 100000 \approx 316.2$ m

Bearings and the atan2 function

In navigation, direction is given as a bearing (or heading), measured clockwise from north. North is $0^{\circ}$ , east $9 0^{\circ}$ , south $18 0^{\circ}$ and west $27 0^{\circ}$ .

If a target lies $Δ N$ metres north and $Δ E$ metres east of you, the bearing to it is

ψ = atan2 (Δ E, Δ N)

Why not simply use $tan^{- 1} (Δ E /Δ N)$ ? Because dividing the two values loses their signs, so you cannot tell north-east from south-west. atan2 takes the two values separately and always returns the angle in the correct quadrant.

Example 2. Bearing to a waypoint

A waypoint is 100 m south ( $Δ N = - 100$ ) and 100 m east ( $Δ E = 100$ ).

$tan^{- 1} (100/ - 100) = tan^{- 1} (- 1) = - 4 5^{\circ}$ , which points north-west: wrong
$atan2 (100, - 100) = 13 5^{\circ}$ , south-east: correct

In Python: math.degrees(math.atan2(dE, dN)) % 360

Ground footprint and GSD

When a camera looks straight down from height $h$ with a given field of view (FOV), the area it captures forms an isosceles triangle, which splits into two right triangles.

Camera at height h looking down with angle FOV covers width W on the ground — Figure 2. A camera at height h and its ground footprint width W

In one half, the adjacent side is $h$ and the opposite side is $W /2$ , so $tan (FOV /2) = \frac{W /2}{h}$ . Rearranging:

W = 2 h tan \frac{FOV}{2}

The ground sampling distance (GSD) is the real ground size covered by one pixel. It equals the footprint width divided by the number of pixels across the same direction:

GSD = \frac{W}{N _{px}}

Example 3. Planning a survey flight

A camera has a horizontal FOV of $8 4^{\circ}$ and images 5472 pixels wide. It flies at 100 m.

$W = 2 \times 100 \times tan 4 2^{\circ} = 200 \times 0.9004 \approx 180.1$ m
$GSD = 180.1 m \div 5472 \approx 0.0329$ m, about 3.3 cm per pixel

At 50 m both $W$ and GSD halve. Images become sharper but you need more flight lines. This is the basic trade-off of mapping.

Resolving wind into components

Wind blowing across the flight path has two effects. The part parallel to the path, the headwind, reduces ground speed. The part perpendicular to it, the crosswind, pushes the drone sideways.

Wind vector V at angle theta to the flight path, split into V cos theta along the path and V sin theta across it — Figure 3. Wind components along and across the flight path

V_{head} = V cos θ V_{cross} = V sin θ

Example 4. Wind at 30 degrees

An 8 m/s wind blows towards the drone’s nose at $3 0^{\circ}$ to the flight path.

Headwind $8 cos 3 0^{\circ} = 8 \times 0.866 \approx 6.9$ m/s
Crosswind $8 sin 3 0^{\circ} = 8 \times 0.5 = 4.0$ m/s

If the drone flies at an airspeed of 15 m/s, its ground speed drops to about $15 - 6.9 = 8.1$ m/s, so the outbound leg takes almost twice as long. In the next module we will calculate this more precisely with vectors.

Common mistakes

Watch out

Not checking calculator mode, DEG versus RAD.
Using the full FOV instead of half. The footprint formula needs $tan (FOV /2)$ .
Using $tan^{- 1}$ for bearings. Always use atan2 to get the right quadrant.
Swapping sin and cos. Name the sides relative to your chosen angle first.

Summary

$18 0^{\circ} = π$ rad, and software computes angles in radians.
Right triangles use Pythagoras and the sin, cos and tan ratios.
Bearings are measured clockwise from north and found with atan2(ΔE, ΔN).
Footprint width is $W = 2 h tan (FOV /2)$ , and GSD is $W$ divided by the pixel count.
Crossing wind splits into headwind $V cos θ$ and crosswind $V sin θ$ .

Check your understanding

Convert $4 5^{\circ}$ and $12 0^{\circ}$ to radians.
A drone is 400 m away horizontally and 120 m up. What is the straight-line distance to it?
A target is 50 m south and 50 m west. What is its bearing?
A camera with a $7 0^{\circ}$ FOV flies at 60 m. What is its footprint width?
A 10 m/s wind is at $6 0^{\circ}$ to the flight path. What is the crosswind?

Answers

$4 5^{\circ} = π /4 \approx 0.785$ rad and $12 0^{\circ} = 2 π /3 \approx 2.094$ rad
$40 0^{2} + 12 0^{2} = 174400 \approx 417.6$ m
$atan2 (- 50, - 50) = - 13 5^{\circ}$ ; adding $36 0^{\circ}$ gives $22 5^{\circ}$ (south-west)
$W = 2 \times 60 \times tan 3 5^{\circ} = 120 \times 0.700 \approx 84.0$ m
$10 sin 6 0^{\circ} \approx 8.66$ m/s

Key formulas

Degrees to radians	$θ_{rad} = θ_{deg} \times \frac{π}{180}$
Pythagoras	$c^{2} = a^{2} + b^{2}$
Trigonometric ratios	$sin θ = \frac{a}{c}, cos θ = \frac{b}{c}, tan θ = \frac{a}{b}$
Bearing from coordinates	$ψ = atan2 (Δ E, Δ N)$
Ground footprint	$W = 2 h tan \frac{FOV}{2}$
GSD	$GSD = \frac{W}{N _{px}}$
Wind components	$V_{head} = V cos θ, V_{cross} = V sin θ$

Key references

Abramson, J. (2021). Algebra and trigonometry (2nd ed.). OpenStax. link
Stroud, K. A., & Booth, D. J. (2020). Engineering mathematics (8th ed.). Bloomsbury.
Wolf, P. R., Dewitt, B. A., & Wilkinson, B. E. (2014). Elements of photogrammetry with applications in GIS (4th ed.). McGraw-Hill.

In class / field

Lecture, case discussion and in-class problem solving

Learning evidence: Quiz results and submitted exercises

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Mathematics, physics and statistics · Control, autopilot and navigation · Mission planning, flight and simulation

Trigonometry and angles in flight

By the end of this module you will be able to

Angles are part of every flight

Degrees and radians

The right triangle

Pythagoras’ theorem

Trigonometric ratios

Bearings and the atan2 function

Ground footprint and GSD

Resolving wind into components

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Trigonometry, Euler angles and reference frames

Coordinates, direction, time and height

In class / field

Module quiz

Trigonometry and angles in flight

Lesson

By the end of this module you will be able to

Angles are part of every flight

Degrees and radians

The right triangle

Pythagoras’ theorem

Trigonometric ratios

Bearings and the atan2 function

Ground footprint and GSD

Resolving wind into components

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Trigonometry, Euler angles and reference frames

Coordinates, direction, time and height

In class / field

Module quiz