Step by step
The question
Question: find side c of the triangle with a = 5, b = 8 and C = 60°.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write the law of cosines
We know two sides, a and b, and the angle C between them.
The law of cosines: c² = a² + b² − 2ab cos C.
Side c is across from angle C.
Where does it come from?
Put corner C at 0, with side a along the x-axis.
The last corner is b times the unit-circle point: b cos C across, b sin C up.
Side c joins two corners.
Pythagoras on the across and up gaps gives c squared.
Open the brackets.
Take b² out of the two square parts.
cos squared plus sin squared is 1.
That leaves the law.
When C is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: a = 5, b = 8 and C = 60°.
3 Work it out
Squares first: 5² is 25, 8² is 64.
And 2 × 5 × 8 is 80.
Careful!
Times comes before take away.
Work out 80 × cos 60° first, then take it away.
The calculator gives cos 60°: exactly 0.5.
80 times that is 40.
25 + 64 is 89.
So c² = 49.
4 Take the square root
c squared is 49.
So c is the square root of 49.
A side is never negative, so only the plus root.
49 is 7 squared.
So c = 7.
5 Check
Let's check: the law turned around gives cos C back.
89 take away 49, over 80: 0.5.
And cos 60° is 0.5.
The same!
So c is right.
6 Pick the choice
Our answer is c = 7.
That is choice A.
7 Why not the others
B is √89.
Wrong: forgot the − 2ab cos C part.
C is √129.
Wrong: added the 2ab cos C part instead of taking it away.
D is 13.
Wrong: 5 + 8: added the sides.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: work it out.
Then: take the square root.
Then: check.
Then: pick the choice.
Last: why not the others.