Step by step
The question
Question: find side a of the triangle with b = 6, c = 9 and A = 120°.
1 Write the law of cosines
We know two sides, b and c, and the angle A between them.
The law of cosines: a² = b² + c² − 2bc cos A.
Side a is across from angle A.
It is Pythagoras with a correction.
When A is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: b = 6, c = 9 and A = 120°.
3 Work it out
Squares first: 6² is 36, 9² is 81.
And 2 × 6 × 9 is 108.
Careful!
Times comes before take away.
Work out 108 × cos 120° first, then take it away.
The calculator gives cos 120°: exactly −0.5.
108 times that is −54.
36 + 81 is 117.
Then we take away −54.
Taking away a negative is adding: − (−54) = + 54.
The angle is more than 90°, so a is longer than Pythagoras gives.
So a² = 171.
4 Take the square root
a squared is 171.
So a is the square root of 171.
A side is never negative, so only the plus root.
The calculator gives about 13.077.
To the nearest tenth, a ≈ 13.1.
5 Check
Let's check: the law turned around gives cos A back.
117 take away 171, over 108: −0.5.
And cos 120° is −0.5.
The same!
So a is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: work it out.
Then: take the square root.
Last: check.