The cosine rule
Watch this video to learn about the cosine rule.
Finding a side
The cosine rule is:
\({a^2} = {b^2} + {c^2} - 2bcCosA\)
Use this formula when given the sizes of two sides and its included angle.
Example
Find the length of BC.
Answer
We have two sides and the included angle.
\({a^2} = {b^2} + {c^2} - 2bcCosA\)
\({a^2} = {7^2} + {3^2} - (2 \times 7 \times 3 \times \cos (35^\circ ))\)
\(a^{2}=49+9-34.40\)
\(a^{2}=23.60\)
\(a=\sqrt{23.60}\)
\(a = 4.9cm\,(to\,1\,d.p.)\)
Now try the example question below.
Question
Find the length of AB.
We have the two sides and the included angle.
\({c^2} = {a^2} + {b^2} - 2ab CosC\)
\({c^2} = {4^2} + {9^2} - 2 \times 4 \times 9 \times Cos27^\circ\)
\(c^2 = 16 + 81 - 64.15\)
\(c^2 = 32.85\)
\(c= \sqrt{32.85}\)
\(c = 5.7cm\)
Finding an angle
An angle in a triangle can be found if you know the size of all the sides.
When this is the case a different version of the cosine rule is used in which the subject has been changed. The forumla is:
\(cosB = \frac{{{{a^2} + {c^2} - {b^2}}}}{2ac}\)
Example
Find the size of the angle AB.
(Notice the pattern in the letters of the formula. Adapt these to suit the question.)
\(cosB = \frac{{{{a^2} + {c^2} - {b^2}}}}{2ac}\)
\(cosB = \frac{{{{4^2} + {5^2} - {7^2}}}}{2 \times 4 \times 5}\)
\(cosB = \frac{{{{16} + {25} - {49}}}}{40}\)
\(cosB = \frac{-8}{40}\)
\(cosB = -0.2\)
\(AngleB = cos{^-}{^1}(-0.2)\)
\(AngleB = 101.5^\circ\)
Question
Find the size of angle R.
As we are calculating the size of an angle, we use the second formula.
\(\cos R = \frac{{{p^2} + {q^2} - {r^2}}}{{2pq}}\)
\(\cos R = \frac{{{{4.2}^2} + {{6.9}^2} - {4^2}}}{{2 \times 4.2 \times 6.9}}\)
\(\cos R = \frac{{49.25}}{{57.96}}\)
\(CosR=0.850\)
\(R=cos^{-1}(0.850)\)
\(R = 31.8^\circ (to\,1\,d.p.)\)
