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Maths Exercices
The law of cosines

The Law of Cosines is a extension of the Pythagorean theorem to use when triangles that are not right-angled.

Consider this triangle:

tseno

Then, the Law of Cosines states that:

tseno
teocos2
teocos3

Proof:

The perpendicular, oc, divides this triangle into two right angled triangles, aco and bco.

First we will find the length of the other two sides of triangle aco in terms of known quantities, using triangle bco.

h = A sin b

Side C is split into two segments, total length C.

ob, length A cos b
ao, length C - A cos b

Now we can use Pythagoras to find B, since B2 = ao2 + h2
\begin{matrix}B^2 & = & (C-A\cos b)^2+A^2\sin^2 b \\ \ &=& C^2-2AC\cos b +A^2\cos^2 b+A^2\sin^2 b\\ \ &=& A^2+C^2-2AC\cos b\end{matrix}

The corresponding expressions for A and C can be proved similarly.

Find x.


c2=a2+b2-2accosC
x2=72+92-2·7·9·cos105º=162.6
x=12.8 cm

Find the missing numbers (round the solution to hundredths):

=

=6

=9

=

=4

=