Step by step
The question
Question: find the area of the triangle with sides 10 and 4, and the angle 150° 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 4 as its longest side.
Its angle at the bottom is 180° − 150° = 30°.
SOH: sin is the opposite over the hypotenuse.
sin 30° = h/4.
To remove the 4 at the bottom, we multiply by 4.
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.
4/4 is 1.
So just h is left.
So h = 4 × sin 30°.
From the unit circle: an angle and 180° minus it have the same sine.
So sin 30° = sin 150°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 10, the height is 4 × sin 150°.
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 10 and 4, and sin 150°.
4 Work it out
10 times 4 is 40.
Half of 40 is 20.
The calculator gives sin 150°: exactly 0.5.
Put the number in.
That is 10.
So the area is 10 square units.
5 Check
Let's check another way: use the side 4 as the base.
Its height is 10 times sin 150°: 5.
A half times 4 times that is 10.
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.