Hello, and welcome to this video on the cosine rule.
This video will cover: What is the cosine rule, When can it be used, How to use the
rule to find both side lengths and angles, and further support.
The cosine rule is to be used for non-right angled triangles.
If the triangle is a right angled triangle, you might consider using Pythagoras or SOH
CAH TOA to solve it.
If the triangle is not a right-angled triangle, consider using the sine rule or cosine rule
to solve it.
The cosine rule is applicable to solving a triangle if there are two side lengths known,
as well as the angle between them – as in these three examples.
In this case, the rule can be used to find the length of the third side.
It can also be used when three side lengths are known, to find the magnitude of any of
the angles in the triangle.
If the sides of a triangle are of fixed length, then there is only one possible triangle that
can be drawn, barring any rotations or reflections of this triangle.
Similarly, if two side lengths are given and the angle between them is stipulated, then
again there is only one possible result for the triangle that can be drawn.
Watch how the angle at the top of this triangle increases and decreases (animation).
As the angle changes, so does the length of the side opposite this angle.
There is a special relationship between the cosine of this changing angle and the length
of its opposite side.
This is known as the cosine rule.
The ratio of the cosine of this angle to the length of the side opposite the angle is constant,
and this is the same for each angle and its opposite side.
Like the sine rule, the cosine rule has two forms, one in which the square of the unknown
side length is the subject, and one in which the cosine of an unknown angle is the subject.
The letters of the angle and its opposite side can be interchanged in both forms of
the equation to calculate any side length or the magnitude of any angle.
In this triangle, there are two known sides and a known angle, but this known angle is
not the included angle between the two known sides.
The other two angles are not known, so the cosine rule cannot be used to find Angle A.
In the second triangle, one of the two sides that form the angle is unknown, even though
the side opposite the angle is known, so the cosine rule cannot be used.
It’s worth noting however, that both of these triangles could be solved using the
sine rule.
In these two triangles the cosine rule can be used.
The 68 degree angle on the left can be deduced and the side length n can be found.
And, on the right, the magnitude of Angle B can be found by using the three known side
lengths.
To find the length n, the appropriate values are substituted into the equation.
Each term is evaluated, and then using the correct order of operations, n squared is
calculated.
The value of n is then found by taking the square root.
To find the magnitude of Angle E, the appropriate values are substituted into the correct form
of the equation to find the cosine of E degrees.
This can be left in fraction form.
The magnitude of E is found by the inverse cosine of this fraction.
For further support, contact your local Academic Skills Numeracy Advisor on your campus.
You can access the Academic Skills numeracy resources on LEO.
You can also search the internet for exercises with solutions on the cosine rule.
In this particular example, there are 199,000 results, so there is no shortage of material
to practice with.