The earlier submit gave a easy and correct approximation for the smaller angle of a proper triangle. Given a proper triangle with sides a, b, and c, the place a is the shortest facet and c is the hypotenuse, the angle reverse facet a is roughly
in radians. The earlier submit labored in levels, however right here we’ll use radians.
If the triangle is indirect slightly than a proper triangle, there an approximation for the angle A that doesn’t require inverse trig capabilities, although it does require sq. roots. The approximation is derived in [1] utilizing the identical collection that’s the foundation of the approximation within the earlier submit, the facility collection for two csc(x) + cot(x).
For an indirect triangle, the approximation is
the place s is the semiperimeter.
For comparability, we are able to discover the precise worth of A utilizing the regulation of cosines.
and so
Right here’s a bit Python script to see how correct the approximation is.
from math import sqrt, acos
def approx(a, b, c):
"approximate the angle reverse a"
s = (a + b + c)/2
return 6*sqrt((s - b)*(s - c)) / (2*sqrt(b*c) + sqrt(s*(s - a)))
def actual(a, b, c):
"actual worth of the angle reverse a"
return acos((b**2 + c**2 - a**2)/(2*b*c))
a, b, c = 6, 7, 12
print( approx(a, b, c) )
print( actual(a, b, c) )
This prints
0.36387538476776243 0.36387760856668505
exhibiting that in our instance the approximation is nice to 5 decimal locations.
[1] H. E. Stelson. Word on the approximate resolution of an indirect triangle with out tables. American Mathematical Month-to-month. Vol 56, No. 2 (February, 1949), pp. 84–95.
