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Arbitrary triangle

It’s possible to find the missing pieces of information of an arbitrary triangle, if three pieces of information are given, assuming that not all of them are angles.

The pieces of information that can be given are the angles A,B,C and the side lengths a,b,c.

If one angle is missing

You know that the angle sum of a triangle always is 1800. You simply have to isolate the angle you what to find in the following equation:

180=A+B+C

The law of sines

There are other ways to determine angles and sides of a triangle. Let’s take a look at the law of sines given by:

frac{a}{sin(A)}=frac{b}{sin(B)}=frac{c}{sin(C)}=2*R

where R is the radius of the circumscribed circle of the triangle (i.e. the circle of which the periphery touches all vertices of the triangle).

If one of the sine formulas is used to find an angle, you will get two solutions, if the angle is less than 90 degrees. The other solution is then given by: 180-v.

You should only use the sine formulas to find an angle, if no other way is possible.

The law of cosines
The relationship between sides and angles of a triangle is also defined in the law of cosines:

cos(A)=frac{b^2+c^2-a^2}{2*b*c}

cos(B)=frac{a^2+c^2-b^2}{2*a*c}

cos(C)=frac{a^2+b^2-c^2}{2*a*b}

The three formulas above can be rewritten, so they are more appropriate to calculate the side lengths:

a^2=b^2+c^2-2*b*c*cos(A)

b^2=a^2+c^2-2*a*c*cos(B)

c^2=a^2+b^2-2*a*b*cos(C)

Trigonometric calculator

Triangle with abc

Angles Side lengths
A: a:
B: b:
C: c: