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Trigonometry for Pilots




Figure: Portraits von Pythagoras, from Wikimedia Commons.

Sum of Angles


Figure: Sum of angles in a triangle, from Eddie's notes.

If you draw a triangle with one leg parallel to a straight line, you discover that the angles along the parallel leg are equal to the angles opposite the angle that touches the straight line, which by definition is an angle of 180°. So you can infer that the sum of the three angles in a triangle is 180°.

If you have a right triangle, you know one of the angles is 90*. That means the sum of the other two angles is 90°.

Sum of Legs


Figure: Right triangle legs, from Eddie's notes.

If two sides of a right triangle are known, the third side can be found using the Pythagorean Theorem:

c2 = a2 + b2

Trigonometric Functions



sin A = side opposite A hypotenuse = a c


cos A = side adjacent A hypotenuse = b c


tan A = side opposite A side adjacent A = a b


cot A = side adjacent A side opposite A = b a


sec A = hypotenuse side adjacent A = c b


csc A = hypotenuse side opposite A = c a

Inverse Trigonometric Functions

If . . . x = sin y . . . then . . . y = arcsin x
If . . . x = cos y . . . then . . . y = arccos x
If . . . x = tan y . . . then . . . y = arctan x
If . . . x = cot y . . . then . . . y = arccot x
If . . . x = sec y . . . then . . . y = arcsec x
If . . . x = csc y . . . then . . . y = arccsc x

A Few Examples

Circling Offset


Figure: 30° Circling offset, from Eddie's notes.

The sine of 30° = 0.5, which offers us easy math for many situations. When approaching a runway to circle to the opposite side, for example, one often offsets 30 degrees to establish a downwind. The distance offset is equal to half the distance covered in the offset leg. More about this: Circling Approach.

Course Deviation


Figure: Course azimuth deviation, from Eddie's notes.

The tangent of an angle provides the relationship of a triangle's two legs adjacent to the right angle. Multiplied by the distance left to travel, the tangent of an angle will provide course deviation. More about this: Stabilized Approach.



Figure: Crosswind chart example, from Eddie's notes.

Just as the sine of 30° = 0.5, the cosine of 60° = 0.5. You can also figure that the sine of 60 or the cosine of 30° is √3 / 2 = 0.87, almost 90 percent. The cosine or sine of 45° is 1 / √2 = 0.71, almost three-fourths. That leads to a few handy rules of thumb when it comes to crosswinds:

  • A thirty degree crosswind is equal to one-half the full wind factor.
  • A forty-five degree crosswind is equal to three-fourths the full wind factor.
  • A sixty degree crosswind is equal to ninety percent of the full wind factor.

More about this: Crosswinds.

Vertical Navigation on Approach


Figure: AGL vs. DME to go, from Eddie's notes.

We often use 300 feet per nautical mile as a wag for how high we should be on final approach. It is pretty close to a three degree glide path, convenient eh?

It is more than convenience, it is trignonometry. The real number, as it turns out, is 318 feet per nautical mile:

Height = 6076 ( tan 3 ° ) = 318feet

More about this: Landing and Approach VVI.

Revision: 20140408