Skinny triangle

A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is particularly simple for skinny triangles that are also isosceles or right triangles: in these cases the need for trigonometric functions or tables can be entirely dispensed with.

The skinny triangle finds uses in surveying, astronomy and shooting.

The approximated solution to the skinny isosceles triangle, referring to figure 1, is;

This is based on the small-angle approximations;

and,

when ${\displaystyle \scriptstyle \theta }$ is in radians.

The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1;

The term ${\displaystyle \scriptstyle {\frac {\pi -\theta }{2}}}$ represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus θ) are equal to π. Applying the small angle approximations to the law of sines above results in;

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