Sine Rule and Cosine Rule


Sine and cosine can be used on any triangle, not just right-angled triangles. But as there is no right angle, there is no hypotenuse, so different formulae are needed

Sine Rule

Notice that the small letters, for the lengths of the sides, are opposite to the corresponding capital letters for the angles.

  So for this triangle…

You are given angles A and C and side c, so use

B=180-(60+45)=75 and now

So, if an angle and the length of its opposite side is known, then the other angles and sides can be worked out, using just one other length or angle. But be careful when the triangle is obtuse angled, because, for example, sin50 = sin130, that is sin x = sin (180-x)

That just leaves the triangle where opposite sides/angles are not given;

Here the Cosine rule is needed.

Having worked this out, you now have side a, opposite to angle A and so the sine rule can be used to work out the other angles.

The cosine rule is also used when the three sides are given.

This method can be used again to find another angle, or the sine rule, which is a bit shorter, can be used.

Using these methods, any triangle can be ‘solved’. That is all of the angles and sides of a triangle can be worked out, if enough information is given to start with.

Of course the sine and cosine rules can be used on right angled triangles, but it easier to use the usual ratios for right angled triangles.

Try out the methods on this question.

Use the cosine rule to work out length a

Then use the sine rule to work out angle B

Finally use C=180-(A+B) to work out C


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