Cosine rule
For any triangle, c^2 = a^2 + b^2 - 2ab cos C. It finds a side when you know two sides and the angle between them.
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Analogy
Like Pythagoras with an extra correction for when the angle is not a right angle.
Example
c^2 = 25 + 49 - 2 x 5 x 7 x cos 60= 74 - 35 = 39c = sqrt(39) = 6.24
In practice
Used to find the straight-line gap between two ships on different bearings.
Practice
Flashcards
- Write the cosine rule.c^2 = a^2 + b^2 - 2ab cos C.
Interview questions
- A triangle has sides 5 cm and 7 cm with a 60 degree angle between them. Find the third side.
- Use the cosine rule
- c^2 = 25 + 49 - 70 cos 60
- cos 60 = 0.5, so c^2 = 39
- c is about 6.24 cm
Questions people ask
When do you use the cosine rule?
Use it when you know two sides and the angle between them, and you want the third side. You can also use it when you know all three sides and want an angle. If you know a matching side and angle pair, use the sine rule.
What happens to the cosine rule when C is 90 degrees?
Since cos 90 = 0, the last part of the formula disappears. You are left with c^2 = a^2 + b^2, which is Pythagoras' theorem. So Pythagoras is a special case of the cosine rule. It is a handy check: right-angled cases should always agree with Pythagoras, and they do.