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Triangle Inequalities

Let a triangle have angles A, B, and C, then inequalities include

sinA+sinB+sinC<=3/2sqrt(3)
(1)
1<=cosA+cosB+cosC<=3/2
(2)
sin(1/2A)sin(1/2B)sin(1/2C)<=1/8
(3)
tan(1/2A)+tan(1/2B)+tan(1/2C)>=sqrt(3)
(4)
cotAcotBcotC<=1/9sqrt(3)
(5)
cotA+cotB+cotC>=sqrt(3)
(6)
(sinAsinBsinC)/(cotA+cotB+cotC)<=3/8
(7)
(tan(1/2A)+tan(1/2B)+tan(1/2C))/(tan(1/2A)tan(1/2B)tan(1/2C))>=9
(8)
cos(1/2A)cos(1/2B)cos(1/2C)>=sinAsinBsinC
(9)
>=sin(2A)sin(2B)sin(2C)
(10)
2<=cos^2(1/2A)+cos^2(1/2B)+cos^2(1/2C)<=9/4
(11)
cot(1/2A)cot(1/2B)+cot(1/2B)cot(1/2C)+cot(1/2C)cot(1/2A)>=9
(12)

(Siddons and Hughes 1929, p. 283), and

(sinA+sinB+sinC)/(cotA+cotB+cotC)<=3/2
(13)

(Weisstein).

See also

Triangle Inequality

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References

Siddons, A. W. and Hughes, R. T. Trigonometry, Parts III-IV. London: Cambridge University Press, 1929.

Referenced on Wolfram|Alpha

Triangle Inequalities

Cite this as:

Weisstein, Eric W. "Triangle Inequalities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleInequalities.html

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