Step by step
The question
Question: find side a of the triangle with A = 45°, B = 75° and c = 10.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side c is across from angle C.
Side a, the one we want, is across from angle A.
2 Find the third angle
We need angle C.
The angles of a triangle add to 180°.
So C is 180° take away the other two.
So C = 60°.
3 Write the law of sines
In every triangle, a side over the sine of its partner angle gives the same number, for all three pairs.
We use the pair we want and the pair we know: a over sin A = c over sin C.
Careful!
Each side goes with the angle across from it, never with an angle next to it.
4 Put the numbers in
Now put the numbers in: A = 45°, c = 10 and C = 60°.
5 Find a
a is on top, over sin 45°.
To remove the sin 45° at the bottom, we multiply by sin 45°.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
sin 45° over sin 45° is 1.
So just a is left.
So a = 10 × sin 45° over sin 60°.
The calculator gives sin 45°: about 0.7071.
And sin 60°: about 0.8660.
Put the numbers in.
That is about 8.165.
To the nearest tenth, a ≈ 8.2.
6 Check
Let's check: both pairs must give the same number.
a over sin 45° is about 11.60.
c over sin 60° is about 11.55.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 45° is less than 60°, and 8.2 is less than 10.
So it is right!
Let's look back.
First: match sides and angles.
Then: find the third angle.
Then: write the law of sines.
Then: put the numbers in.
Then: find a.
Last: check.