Two sides of a triangle have lengths a and b and the angle between them is θ. What value of θ will maximize the area of the triangle?

Given,

• The length two sides of a triangle are ‘a’ and ‘b’

• Angle between the sides ‘a’ and ‘b’ is θ.

• The area of the triangle is maximum.

Let us consider,

The area of the ΔPQR is given be

---- (1)

For finding the maximum/ minimum of given function, we can find it by differentiating it with θ and then equating it to zero. This is because if the function A (θ) has a maximum/minimum at a point c then A’(c) = 0.

Differentiating the equation (1) with respect to θ:

---- (2)

[Since ]

To find the critical point, we need to equate equation (2) to zero.

Cos θ = 0

Now to check if this critical point will determine the maximum area, we need to check with second differential which needs to be negative.

Consider differentiating the equation (2) with θ :

----- (2)

[Since ]

Now let us find the value of

As , so the function A is maximum at

As the area of the triangle is maximum when