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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.
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