Law of Cosines

The Law of Cosines is a method used to solve any triangle, given only a side, an angle, and another side.





It is derived using basic trigonometric identities.

The first step is simplifying c and h:

sinA=

h=c(sinA) b=c(cosA)


Then you use pythagorean theorem on the smaller triangle

a^2=h^2 + (b-x)^2

a^2=[c(sinA)]^2 + [b-c(cosA)]^2

a^2=c^2(sinA)^2 + b^2 - 2bc(cosA) + c^2(cosA)^2

a^2=b^2 + c^2 - 2bc(cosA)

The simplified formula is the standard form of the law of cosines.

There are 3 variations of the standard form of the law of cosines, one for each of the 3 variables involved; a, b, and c.

The only difference between them is which variable is used in the location where a is in the formula above.

The variable that goes there is the one of which you want to know the cosine of. The b and c variables are the other two variables in the triangle.

There is a second form of the Law of Cosines. It is called the alternative form, go figure.

cosA =

There are 3 variations of the alternative form as well, which follow the same rules as the variations for the standard form.


Heron's Formula is a method to find the area of a triangle, knowing just the sides.

Area=

s stands for semi-perimeter, which is, as implied, half of the perimeter;

s=

so the full version of the formula comes out to:

Area =

Dylan Dulberg