Step by step
The question
Question: find the area of the triangle with sides 20 and 3, and the angle 100° between them.
1 Find the height
The area needs a height.
Draw it from the top corner straight down to the bottom line.
The angle is more than 90°, so it lands outside the triangle.
The small right triangle outside has the side 3 as its longest side.
Its angle at the bottom is 180° − 100° = 80°.
SOH: sin is the opposite over the hypotenuse.
sin 80° = h/3.
To remove the 3 at the bottom, we multiply by 3.
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.
3/3 is 1.
So just h is left.
So h = 3 × sin 80°.
From the unit circle: an angle and 180° minus it have the same sine.
So sin 80° = sin 100°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 20, the height is 3 × sin 100°.
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 20 and 3, and sin 100°.
4 Work it out
20 times 3 is 60.
Half of 60 is 30.
The calculator gives sin 100°: about 0.9848.
Put the number in.
That is about 29.544.
To the nearest tenth, the area is about 29.5 square units.
5 Check
Let's check another way: use the side 3 as the base.
Its height is 20 times sin 100°: about 19.7.
A half times 3 times that is about 29.5.
The same area!
So it is right.
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.