If you’re just joining us for this Pilot Knowledge series on The Training Aviator, you might want to read the first part of Wind, Navigation, and You before proceeding, unless your grasp of basic meteorology and airspeed definitions is strong. No points taken against you to refer back to that article in any event.
The Wind Triangle
The wind triangle is the graphical representation of the relationship between the airplane’s motion and the wind. To build one we need to know the airplane’s true heading, the airplane’s true airspeed, the wind direction (again, true), and the wind speed.
A wind triangle consists of three vectors:
- True track (path over the ground)
- Wind vector (speed and direction of the wind)
- True heading and true airspeed (the aircraft’s motion through the air)
By constructing this triangle, we can solve for two critical values:
- Wind correction angle (WCA)
- Groundspeed (GS)
Imagine that we have an airplane with a true airspeed of 100 knots, and wind from 360° True at 20 knots. If we want to fly a course of 090° True, what would we need for a wind correction angle, and what would we get for a groundspeed?
From there, we’ll draw our wind vector. Put the tail of the wind vector 100 knots from the tail of the course line, with the tip of the wind vector touching the course line.
The angle of the “100 knots” line tells us the heading we need to fly (and the difference between that and the course line is our wind correction angle). The distance between the tail of the course line and the tip of the wind vector is our ground speed.
Solving the Wind Triangle With Trigonometry
For an easier, faster, and more precise solution, the wind triangle can also be solved analytically using trigonometry. Let’s start with what we know:
- True Airspeed (TAS) is 100 kt
- Wind Speed (WS) is 20 kt
- The Wind Direction (WD) is 360° (or 0°, depending on which is more convenient).
- Desired Course (TC) is 090°
The first thing you need to do is calculate the angle between the wind direction and our ground track. In this case, 90°. This angle is normally represented with the Greek letter 𝜃 (pronounced “Theta”).
Degrees or Radians
Before you start doing trigonometry on your calculator, remember that there are two units for angles: degrees and radians. Make sure you put your calculator into the “degrees” mode. Every calculator is a little different and you may need to refer to the manual, but here is how to do it on an iPhone (at least as of this writing).
On the left side of the figure below, you’ll see that there is an annunciation “RAD” in the bottom left corner of the “screen” on the calculator. On the right side, you’ll see that the button directly above the “division” symbol is labeled “Deg.” These two indications tell us that the calculator is in radians mode.
If you push the button that says “Deg”, you’ll notice that the “Rad” annunciation disappears, and the button changes to “Rad”. We are now in degrees mode.
Radians is the default mode for pretty much every calculator. It’s not hard to convert between radians and degrees (there are 2𝜋 radians in a 360° circle). But, as every airplane compass is calibrated in degrees, it’s a lot easier to just hit the “Deg” button.
Calculating the Wind Correction Angle
The equation for calculating your wind correction angle is:
To work through this, we need to follow the PEMDAS order of operations. That is Parentheses, Exponents, Multiplication & Division (left to right), and Addition & Subtraction (left to right).
- With your calculator in the “Degrees” mode, find the sine of 90°. (Enter “90” and press the sin key). The result should be 1.
- Then multiply the result by the wind speed (20 kt) and divide by the true airspeed (100 kt). Make sure that your speeds are both in the same unit (kt, mph, km/h, etc.). The result should be 0.2.
- Finally, find the arcsine of the results. To do this, simply press the 𝒔𝒊𝒏-𝟏 or arcsin button (whichever one your calculator has). The result should be about 11.53°.
This tells us we need to turn 11.53°, but it doesn’t tell us which way. Just turn into the wind.
Calculating Your Groundspeed
To calculate your groundspeed, use the following equation:
Again, we’ll follow PEMDAS with our calculator in the degrees mode. The process goes like this.
- Start by calculating the crosswind component by finding the cosine of the wind correction angle. Start with the wind correction angle (11.53°and use the cos button. The result should be about 0.9798. Write that down.
- Next, we’ll find the headwind / tailwind component by calculating the cosine of 𝜃 (90°) and multiplying that with the wind speed (20 kt). We should get 0.
- Multiply the true airspeed (100 kt) with the crosswind component (0.9798) and then subtract the headwind component (0). We should find that we have a ground speed of about 97.98 kt.
That takes us to the close of Part 2. Join us next week for Part 3, in which we’ll show you how to do all of this using a traditional E6-B flight computer.
If you have a heading of 090 and a wind FROM 360, your ground track will be to the South of 090, not North of it.
Yes, this confused me as well. And the green “ground track” vector is longer than the heading vector, despite the lower speed, which also makes no sense.
Yeah, this is pretty simple geometry because the example is a right triangle. So cSQR = aSQR + bSQR. Ground speed = SQRT(120*120 + 20*20). = 121.655, slightly faster than the TAS because the wind gives a very slight tailwind component. Then to find wind correction angle, note that its tangent is 20/120=1/6. WCA = ARCTAN(1/6) = 9.462 degrees. So the ground track is 99.462 degrees.
Thank you for the catch—we’re working on updating the images and will republish when it has been corrected.
Reading this makes me appreciate the elegance of an E6B; a pencil mark, a few turns of the wheel and viola! Ground speed and wind correction angle.
Yes! That’s how I learned to fly in the Air Force. We calculated everything on the E6-B. Does anyone remember ICE-T?
After becoming a “real” pilot, I took up the TLAR method (that looks about right). It takes a lot less time to pick a heading, fly for a minute, take note of the ADF or VOR needle (remember the RMI?), make a correction, and repeat. After at most three cycles, you’re spot on the proper heading to maintain the track. Ah… The old days!
A lot easier if you just remember trig for 30 and 60 degrees…30 degree xwind is essentially full windspeed as headwind/tailwind (~86%, close enough) and half amount as xwind, 60 degree xwind is those values reversed.
If it keeps you up at night…45 degrees is about 70% for both…no need to make it difficult or require a calculator, more important to look outside than spend time calculating last few knots!
As far as “true” heading or “true” course in discussion, true or magnetic isn’t important as long as heading/course you’re using are both true or both mag.
This is gibberish to me. I didn’t know that I had to be a math major to fly an airplane. I never made it past high school plane geometry. Thank goodness for the E6-b.