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Maths Exercices
Law of sines

The Law of Sines is the relationship between sides and angles in any triangle.

The sides of a triangle are to one another in the same ratio as the sines of their opposite angles.

Let's see...

Consider this triangle:

tseno

Then, the Law of Sines states that:

tseno

The law can also be written as the reciprocal:

tseno

Proof:

The perpendicular, oc, splits this triangle into two right-angled triangles. This lets us calculate h in two different ways

  • Using the triangle cao gives tseno
  • Using the triangle cbo gives h=A \sin b \,
  • Eliminate h from these two equations A \sin b =B \sin a \,
  • Rearrange {A \over \sin a} = {B \over \sin b}

By using the other two perpendiculars the full law of sines can be proved. QED.


Find AC.


= ; AC·sin50=10·sin88; AC= ; AC=13 cm


Complete the following triangle using the law of sines: (Round the solution to hundredths)

=

=8

=9

=45

=

=