Step by step
The question
Question: find side a of the triangle with b = 7 cm, c = 5 cm and A = 70°.
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 = 7, c = 5 and A = 70°.
3 Work it out
Squares first: 7² is 49, 5² is 25.
And 2 × 7 × 5 is 70.
Careful!
Times comes before take away.
Work out 70 × cos 70° first, then take it away.
The calculator gives cos 70°: about 0.3420.
70 times that is about 23.94.
49 + 25 is 74.
So a² ≈ 50.06.
4 Take the square root
a squared is 50.06.
So a is the square root of 50.06.
A side is never negative, so only the plus root.
The calculator gives about 7.075.
To the nearest tenth, a ≈ 7.1 cm.
5 Check
Let's check: the law turned around gives cos A back.
74 take away 50.06, over 70: 0.342.
And cos 70° is about 0.3420.
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.