Small Angle Approximations and Skinny Triangles are concepts which simplify trigonometric calculations. The concept is based on right-angle or isosceles triangles where one side is very small compared to the other two. In these cases, some approximations can be made in place of trigonometric calculations, which may be used for estimations when information required to compute an exact value is missing, or where an estimated value is good enough.

Consider the following triangle:

Small Angle Approximation - 45 degree triangle

On this right-angle triangle, the blue arc a represents the the edge of a circle if the shape was a slice of a full circle (ie: the shape bounded by the green, blue and bottom black lines would be a portion of a circle). From this, we see that r represents the radius of the circle, y is the triangle height and h is the hypotenuse. Θ is the angle from the middle of the circle, which we represent in radians for this approximation.

When Θ = π/4 (0.785 radians / 45°) as shown above, we can’t see any obvious relationship between these values:
when r = 100:
h = 141.4
y = 100
a = 78.5
sinΘ = 0.707
tanΘ = 1
cosΘ = 0.707
1 – (Θ²/2) = 0.692

But watch what happens as Θ decreases (all triangles drawn to scale).

Small Angle Approximation - 26.6 degree right angle triangle

Θ = 0.464 radians (26.6°)
when r = 100:
h = 111.8
y = 50
a = 46.4
sinΘ = 0.448
tanΘ = 0.500
cosΘ = 0.894
1 – (Θ²/2) = 0.892

Small Angle Approximation - 14.0 degree right-angle triangle

Θ = 0.244 radians (14.0°)
when r = 100:
h = 103.1
y = 25
a = 24.4
sinΘ = 0.242
tanΘ = 0.249
cosΘ = 0.970
1 – (Θ²/2) = 0.970

As you can see, for small values of Θ:

  • rh
  • ya
  • sinΘtanΘΘ
  • cosΘ ≈ 1 – (Θ²/2)

As Θ approaches zero, the errors in these approximations also approach zero. As you can see, Θ doesn’t need to be that small for these approximations to be accurate enough for quick calculations. These approximations are used in a number of fields, as diverse as astronomy, optics and aircraft navigation to name a few.

  1. Laeeq Khan says:

    Nice to see some people still appreciate beauty of mathematics. It is sad that, in this age of fancy calculators/iPhones most high schoolers have no capacity of comprehending what you are describing here.