1. Starter Sketch a regular pentagon
Join each vertex to every other vertex with a straight line
How many different sized triangles can you find?

2. We are Learning to……
Use The Cosine Law

3. The cosine law Consider any triangle ABC.
If we drop a perpendicular line, h from C to AB, we can divide the triangle into two right-angled triangles, ACD and BDC. C b a h A B
a is the side opposite A and b is the side opposite B. x D
c – x
We call the perpendicular h for height (not h for hypotenuse).
c is the side opposite C. If we call the length AD x, then the length BD can be written as c – x.

4. The cosine law Using Pythagoras’ theorem in triangle ACD, C
b2 = x2 + h2 1 b a h
Also,
cos A = x b A B x D
c – x
x = b cos A 2
In triangle BCD,
a2 = (c – x)2 + h2
We call the perpendicular h for height (not h for hypotenuse).
To derive the cosine rule we use both Pythagoras’ Theorem and the cosine ratio.
a2 = c2 – 2cx + x2 + h2
a2 = c2 – 2cx + x2 + h2
Substituting and , 1 2
This is the cosine law.
a2 = c2 – 2cb cos A + b2
a2 = b2 + c2 – 2bc cos A

5. The cosine law For any triangle ABC, A B C c a b
a2 = b2 + c2 – 2bc cos A or
cos A =
b2 + c2 – a2
2bc
We can use the first form of the formula to find side lengths and the second form of the equation to find angles.

6. Using the cosine law to find side lengths
If we are given the length of two sides in a triangle and the size of the angle between them, we can use the cosine law to find the length of the other side. For example,
Find the length of side a. B C A
7 cm
4 cm
48° a
a2 = b2 + c2 – 2bc cos A
The angle between two sides is often called the included angle.
We can express the cosine rule as, “the square of the unknown side is equal to the sum of the squares of the other two sides minus 2 times the product of the other two sides and the cosine of the included angle”.
Warn pupils not to forget to find the square root. The answer should look sensible considering the other lengths.
Advise pupils to keep the value for a2 on their calculator displays. They should square root this value rather than the rounded value that has been written down. This will avoid possible errors in rounding.
Point out that if we are given the lengths of two sides and the size of an angle that is not the included angle, we can still use the cosine rule to find the length of the other side. In this case we can either rearrange the formula or substitute the given values and solve an equation.
a2 = – 2 × 7 × 4 × cos 48°
a2 = (to 2 d.p.)
a = 5.25 cm (to 2 d.p.)

7. Using the cosine law to find side lengths
Be aware that the lengths of the sides and the angles have been rounded. This means that, for example, the three angles in the triangle may not add up to exactly 180°.
Change the shape of the triangle by dragging on the vertices.
Reveal the lengths of two of the sides and the angle between them. Ask a volunteer to show how the cosine rule can be used to find the length of the third side.
Make the problem more difficult by revealing two sides and an angle other than the one between them.

8. Using the cosine law to find angles
If we are given the lengths of all three sides in a triangle, we can use the cosine law to find the size of any one of the angles in the triangle. For example,
Find the size of the angle at A.
4 cm
8 cm
6 cm A B C
cos A =
b2 + c2 – a2
2bc
cos A =
– 82
2 × 4 × 6
Point out that if the cosine of an angle is negative, we expect the angle to be obtuse. This is because the cosine of angles in the second quadrant is negative. We do not have the same ambiguity as with the sine rule where the sine of angles in both the first and second quadrants are positive and so two solutions between 0° and 180° exist. Angles in a triangle can only be within this range.
This is negative so A must be obtuse.
cos A = –0.25
A = cos–1 –0.25
A = ° (to 2 d.p.)

9. Using the cosine law to find angles
Be aware that the lengths of the sides and the angles have been rounded. This means that, for example, the three angles in the triangle may not add up to exactly 180°.
Change the shape of the triangle by dragging on the vertices.
Reveal the lengths of all three sides. Ask a volunteer to show how the cosine rule can be used to find the size of a required angle.

10. McGraw Hill Page 110 #1 – 6 BLM 2.8 #s 1 – 4
To succeed at this lesson today you need to…
1. Use this when the problem involves three sides and an angle
2. It works for all types of triangle
3. The law is: a² = b² + c² - 2bcCOSA
McGraw Hill Page 110 #1 – 6 BLM 2.8 #s 1 – 4

11. Homework
McGraw Hill Page 110 #s 7 – 11
BLM 2.8 #s 5 – 8