Why heading and course are different things
An aeroplane flies through the air, and the air moves over the ground. Those two motions add as vectors, so the direction the nose points — the heading — is almost never the direction the aircraft actually travels over the ground, which is the course or track. The wind correction angle is the difference between them, and flying it is what stops a 90-minute leg finishing thirty miles downwind of where you intended.
Pilots call the manoeuvre crabbing, because from the ground the aircraft appears to move sideways like a crab. It is not a manoeuvre at all in the air-mass frame: the aircraft is flying perfectly straight through a body of air that happens to be sliding across the earth. That distinction is the whole content of the wind triangle, and it explains why the correction depends on the ratio of wind speed to airspeed rather than on wind speed alone. A 20 kt crosswind is an 11.5 degree problem at 100 kt and a 2.7 degree problem at 430 kt.
The same vector sum also produces the ground speed, and the two answers are not independent. Whenever you crab into a crosswind, part of your airspeed is spent fighting sideways instead of pushing you along the course, so the cosine term in the ground speed equation always removes something. That loss is small — at 11.5 degrees of crab you keep 98% of your airspeed along the track — but it is never a gain.
The wind triangle, term by term
Draw the triangle with three sides. The wind vector runs from the origin in the direction the wind is blowing towards, with length equal to wind speed. The true airspeed vector runs along the heading with length equal to true airspeed. The sum of those two is the ground vector, which must lie along the course you want.
Start by reducing the wind direction to an angle relative to the course. Call it A = wind direction − true course, taken between −180° and +180°. Then two components fall out immediately:
- Crosswind component = Vw · sin A. Positive here means the wind comes from the right of the course.
- Headwind component = Vw · cos A. Positive is a headwind; negative is a tailwind.
The crosswind component must be exactly cancelled by the sideways part of the airspeed vector, and the sideways part of the airspeed vector is V · sin(WCA). Setting them equal and solving gives WCA = arcsin(Vw · sin A / V). That is the law of sines, and it is the entire derivation. The heading is then simply the course plus the correction angle, with a positive correction meaning a turn to the right — into a wind from the right.
Ground speed is what is left of the airspeed once the sideways part has been used up, minus the along-course wind: GS = V · cos(WCA) − Vw · cos A. Notice that the headwind term enters with a minus sign, so a tailwind — negative cos A — contributes positively. Both terms matter, and neither dominates by right: a wind just abaft the beam gives a small tailwind and a large crab, and the cosine loss can exceed the tailwind entirely. A 50 kt wind 95° off the course at 100 kt TAS is a 4.4 kt tailwind, a 29.9° crab and a ground speed of 91.1 kt — below true airspeed despite the tailwind. Pilots who subtract the headwind component and stop there over-estimate ground speed on any strongly crosswind leg.
When the crosswind component exceeds true airspeed, the arcsine has no real value and the equation is telling you something physical: no heading holds that course, and the aircraft is displaced downwind no matter what you do. Balloonists and slow-flying aircraft in strong winds meet this regularly.
Worked example: true course 040°, TAS 120 kt, wind 300° at 20 kt
You are planning a leg with a true course of 040°, cruising at 120 kt true airspeed. The forecast wind aloft is 300° at 20 kt.
- Wind angle relative to the course. 300 − 40 = 260°, which is more than 180, so subtract a full circle: A = −100°. The negative sign means the wind is from the left.
- Crosswind component. sin(−100°) = −0.98481, so the crosswind is 20 × −0.98481 = −19.70 kt, that is 19.7 kt from the left.
- Headwind component. cos(−100°) = −0.17365, so the along-course component is 20 × −0.17365 = −3.47 kt — a small tailwind.
- Correction angle. WCA = arcsin(−19.70 ÷ 120) = arcsin(−0.16413) = −9.45°. Negative, so the heading lies left of the course, into the wind.
- True heading. 040 − 9.45 = 030.6°, which you would fly as 031.
- Ground speed. GS = 120 × cos(9.45°) − (−3.47) = 120 × 0.98644 + 3.47 = 118.37 + 3.47 = 121.85 kt.
- Time for a 150 NM leg. 150 ÷ 121.85 × 60 = 73.9 minutes.
Sanity check the shape of the answer before you accept it. The wind is from the left, so the heading must be to the left of the course — it is. The along-course component is a small tailwind, so ground speed should be slightly above true airspeed — 121.85 against 120, and the cosine loss of 1.63 kt has eaten most of the 3.47 kt tailwind. That last detail is exactly what pilots miss when they estimate ground speed by subtracting the headwind component alone.
Reading the numbers, and the shortcuts worth knowing
The correction angle is the number you fly; the ground speed is the number you plan with. Round the heading to the nearest degree, because no directional gyro or heading bug resolves better than that, and a tenth of a degree over a 100 NM leg is 0.17 NM, about 320 metres of displacement.
Two mental approximations get you close enough to check the calculator. First, for small angles arcsin(x) in degrees is roughly 60x, so WCA ≈ 60 × crosswind ÷ TAS. With a 19.7 kt crosswind at 120 kt that gives 9.85° against the exact 9.45° — good to half a degree. Second, the crosswind component itself follows the clock rule: a wind 15° off the course gives about a quarter of its speed as crosswind, 30° gives a half, 45° gives 70%, 60° gives 87%, and 90° gives all of it. The reference table below lists those factors exactly.
When the correction angle passes about 25 degrees the leg deserves a second look. That only happens when the crosswind is more than 40% of your true airspeed, and it means both a big heading offset and a substantial cosine loss on ground speed. In light aircraft this regime usually appears in mountain wave or jet-stream conditions, and the correct response is often a different altitude rather than a bigger crab.
Do not confuse this crosswind component with the one you use for landing. Both come from the same sine relationship, but the runway calculation uses the surface wind and the runway direction, and it is compared against the aircraft's demonstrated crosswind limit rather than converted into a heading. The crosswind component calculator is the right tool for that.
Wind components and correction angle by wind angle
| Wind angle off course | Crosswind factor (sin) | Headwind factor (cos) | Crosswind (kt) | WCA (deg) | Ground speed (kt) |
|---|---|---|---|---|---|
| 0° (dead ahead) | 0.000 | 1.000 | 0.0 | 0.00 | 80.0 |
| 15° | 0.259 | 0.966 | 5.2 | 2.97 | 80.5 |
| 30° | 0.500 | 0.866 | 10.0 | 5.74 | 82.2 |
| 45° | 0.707 | 0.707 | 14.1 | 8.13 | 84.9 |
| 60° | 0.866 | 0.500 | 17.3 | 9.97 | 88.5 |
| 75° | 0.966 | 0.259 | 19.3 | 11.14 | 92.9 |
| 90° | 1.000 | 0.000 | 20.0 | 11.54 | 98.0 |
| 120° | 0.866 | −0.500 | 17.3 | 9.97 | 108.5 |
| 150° | 0.500 | −0.866 | 10.0 | 5.74 | 116.8 |
| 180° (dead astern) | 0.000 | −1.000 | 0.0 | 0.00 | 120.0 |
Angles are measured from the course to the direction the wind blows from. The sin and cos factors are universal; the knot and degree columns are those factors evaluated at 20 kt of wind and 100 kt of true airspeed.
Mistakes that make a wind triangle answer wrong
- Mixing true and magnetic references. Forecast winds aloft are given in degrees true; tower and ATIS surface winds are given in degrees magnetic. Solve the triangle entirely in true, then apply variation once at the end with the magnetic variation calculator.
- Using indicated airspeed instead of true airspeed. At 8,000 ft on a standard day, indicated airspeed understates true airspeed by roughly 13%, and the wind triangle needs the true figure. Convert with the true airspeed calculator first.
- Applying the correction the wrong way. The heading always turns towards the wind, never away. A wind from the right gives a heading right of the course. If your answer points downwind, you have inverted a sign.
- Subtracting the headwind component and stopping there. That ignores the cosine loss from crabbing. With a 30 kt wind 60 degrees off the course at 120 kt, the crosswind is 25.98 kt, the crab is 12.50 degrees, and the cosine loss is 120 × (1 − 0.9763) = 2.8 kt — minutes over a long leg. The loss grows as the square of the crab angle, so it is negligible below 10 degrees and severe beyond 25.
- Treating the wind as constant along the route. Wind veers and strengthens with height and changes across a frontal zone. Break a long leg at the points where the forecast changes and solve each segment.
- Forgetting that the correction is a ratio. The same wind produces a much larger angle in a slow aircraft. Doubling true airspeed roughly halves the correction angle for the same crosswind.
The reverse problem: finding the wind from your track
If you already know your heading, true airspeed, actual track and actual ground speed, the same triangle can be solved backwards to recover the wind. Subtract the ground vector from the air vector and the remainder is the wind. In practice most pilots do it approximately: the drift angle you are holding tells you the crosswind component directly, since crosswind ≈ TAS × sin(drift), and the difference between ground speed and true airspeed tells you most of the along-course component. Two legs on different headings give enough information to pin down both the direction and the speed.
Where the wind triangle sits in flight planning
The wind triangle is the oldest computation in air navigation and it is still the one every flight plan rests on. Manual flight computers — the circular E6B and the slide-type CR series — exist mainly to solve it, and they do so graphically rather than trigonometrically, which is why their answers agree with this page to within the width of a pencil line. Modern flight management systems solve the same triangle continuously, updating the wind from the difference between the inertial ground vector and the air data.
In practice you solve it three times for a single flight. Once during planning, from the forecast winds aloft, to get headings and leg times that feed the flight time and ETA calculator and from there the fuel plan. Once in the air, where the observed drift and ground speed tell you what the wind is really doing and let you revise the plan. And once on arrival, where the surface wind becomes the runway crosswind problem instead.
Marine navigation runs the identical mathematics under different names. A vessel setting a course through a current solves the same triangle, calling the correction the set and drift problem, and the current takes the place of the wind. If you navigate a boat as well as fly, the set and drift calculator is the same triangle with the labels changed.
Key terms
- True course (TC)
- The direction of the intended track over the ground, measured clockwise from true north.
- True heading (TH)
- The direction the aircraft's nose points, measured from true north. Equal to the true course plus the wind correction angle.
- Wind correction angle (WCA)
- The angle between heading and course, applied into the wind. Also called the crab angle.
- Drift angle
- The angle between heading and the track actually made good. When you are correcting perfectly, the drift angle equals the wind correction angle with the opposite sign.
- Crosswind component
- The part of the wind acting perpendicular to the course, equal to wind speed multiplied by the sine of the angle between them.
- Variation
- The local angle between true north and magnetic north. Subtract easterly variation from a true heading to get the magnetic heading.