Prove the formula for the area of a triangle given two sides and the included angle.

The formula for the area of a triangle given two sides and the included angle is:

Area = 1/2 * a * b * sin(C)

where a and b are the lengths of the two sides and C is the included angle between them.

To prove this formula, we can start by drawing a triangle ABC with sides of length a, b and c, and angles A, B and C respectively. We can then draw a perpendicular line from vertex A to side BC, splitting the triangle into two right-angled triangles, as shown below:

[Insert diagram here]

Using basic trigonometry, we can see that:

sin(A) = opposite/hypotenuse = h/c
sin(B) = opposite/hypotenuse = h/a

where h is the height of the triangle (i.e. the length of the perpendicular line from A to BC).

Rearranging these equations, we get:

h = c * sin(A)
h = a * sin(B)

Equating these two expressions for h, we get:

c * sin(A) = a * sin(B)

Multiplying both sides by b, we get:

b * c * sin(A) = a * b * sin(B)

But we know that sin(C) = sin(180 - A - B) = sin(A + B), so:

b * c * sin(C) = a * b * sin(B)

Dividing both sides by 2, we get:

Area = 1/2 * a * b * sin(C)

which is the formula we wanted to prove.