Coaching Centre
The cosine rule
The cosine rule for finding a side from two sides and the included angle, and finding an angle from three sides.
The rule
When C = 90°, cos C = 0 and the cosine rule becomes Pythagoras' theorem.
$a = 7$, $b = 9$, $C = 60^\circ$. Find $c$.
A triangle has sides $5$, $6$ and $7$. Find the largest angle.
$a = 7$, $b = 9$, $C = 60^\circ$. Find $c$ (2 d.p.).
$\sqrt{67}$.
Sides $5$, $6$, $7$. Largest angle (1 d.p.)?
$\cos^{-1}(0.2)$.
$a = 10$, $b = 12$, $C = 110^\circ$. Find $c$ (2 d.p.).
$c^2 = 244 - 240\cos 110^\circ$.
Sides $8$, $10$, $12$. Angle opposite $12$ (1 d.p.)?
$\cos C = \tfrac{64 + 100 - 144}{160} = 0.125$.
$a = 5$, $b = 8$, $C = 60^\circ$. Find $c$.
$25 + 64 - 40 = 49$.
Which suits the cosine rule?
No opposite pair is known.
Sides $3$, $5$, $7$. Angle opposite $7$?
$\cos C = \tfrac{9 + 25 - 49}{30} = -0.5$.
$a = 5$, $b = 12$, $C = 90^\circ$. Find $c$.
Pythagoras.
New numbers every time. Keep going until you get five in a row.