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Triangle.Scalene

An example of a scalene triangle.

Scalene triangles are defined as a triangle where the interior angles are all different. Most triangles drawn at random would be scalene. The converse of this is also true - If all three angles are different, then the triangle is scalene, and all the sides are different lengths.

Area

The area of a scalene triangle is:

A= \sqrt{\frac{(a^2+b^2+c^2)^2}{16}-\frac{a^4+b^4+c^4}{8}}

See the heron's formula for more information.

Perimeter

The perimeter of a scalene triangle is

P= a+b+c

Trigonometric functions of half angles in a triangle

Sin= \sqrt{\frac{a^2}{4bc}-\frac{b}{4c}-\frac{c}{4b}+\frac{1}{2}}
Cos= \sqrt{\frac{b}{4c}-\frac{a^2}{4bc}+\frac{c}{4b}+\frac{1}{2}}
Tan= \sqrt{\frac{a^2-b^2+2bc-c^2}{b^2-a^2+2bc+c^2}}
Csc= \sqrt{\frac{4bc}{a^2-b^2+2bc-c^2}}
Sec= \sqrt{\frac{4bc}{b^2-a^2+c^2+2bc}}
Cot= \sqrt{\frac{b^2-a^2+2bc+c^2}{a^2-b^2+2bc-c^2}}

Trigonometric functions of whole angles in a triangle

Sin= \frac{opposite}{hypotenuse}=\frac{2A}{bc}= \sqrt{\frac{(a^2+b^2+c^2)^2}{4b^2c^2}-\frac{a^4+b^4+c^4}{2b^2c^2}}
Cos= \frac{adjacent}{hypotenuse}=\frac{b}{2c}+\frac{c}{2b}-\frac{a^2}{2bc}
Tan= \frac{opposite}{adjacent}=\frac{4A}{b^2+c^2-a^2}=\sqrt{\frac{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)}{(b^2+c^2-a^2)^2}}
Csc= \frac{hypotenuse}{opposite}=\frac{bc}{2A}=\sqrt{\frac{4b^2c^2}{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}
Sec= \frac{hypotenuse}{adjacent}= \frac{2bc}{b^2+c^2-a^2}
Cot= \frac{adjacent}{opposite}=\frac{b^2+c^2-a^2}{4A}=\sqrt{\frac{(b^2+c^2-a^2)^2}{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)}}

CircumCircle

R= \frac{abc}{\sqrt{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)}}

InCircle

r= \frac{\sqrt{\frac{(a^2+b^2+c^2)^2}{4}- \frac{a^4+b^4+c^4}{2}}}{a+b+c}

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