Step by step
The question
Question: find the area of the triangle with sides 8 and 11, and the angle 50° between them.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the height
The area needs a height.
Draw it from the top corner straight down to the base, at a square corner: h.
The small right triangle on the left has the side 11 as its longest side, and h across from the 50° angle.
SOH: sin is the opposite over the hypotenuse.
sin 50° = h/11.
To remove the 11 at the bottom, we multiply by 11.
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.
11/11 is 1.
So just h is left.
So h = 11 × sin 50°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 8, the height is 11 × sin 50°.
So the rule is ½ × a × b × sin C: two sides and the sine of the angle between them.
Careful!
The angle must be the one between the two sides we use.
Another angle gives a wrong height.
3 Put the numbers in
Now put the numbers in: the two sides 8 and 11, and sin 50°.
4 Work it out
8 times 11 is 88.
Half of 88 is 44.
The calculator gives sin 50°: about 0.7660.
Put the number in.
That is about 33.706.
To the nearest tenth, the area is about 33.7 square units.
5 Check
Let's check another way: use the side 11 as the base.
Its height is 8 times sin 50°: about 6.13.
A half times 11 times that is about 33.7.
The same area!
So it is right.
6 Pick the choice
Our answer is A ≈ 33.7.
That is choice B.
7 Why not the others
A is 28.3.
Wrong: used cos 50°: the area uses sin.
C is 44.
Wrong: ½ × 8 × 11: left out sin C.
D is 67.4.
Wrong: forgot the ½.
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Then: check.
Then: pick the choice.
Last: why not the others.