Step 1: Write the formula for area using two sides and the included angle.
If two sides of a triangle are $a$ and $b$ with included angle $\theta$, then: \[A = \frac{1}{2} ab \sin\theta\] This comes from: area = half times base times height, and height = $b\sin\theta$.
Step 2: Identify the fixed and variable quantities.
Since $a$ and $b$ are the given fixed sides, the factor $\frac{1}{2}ab$ is a positive constant. The only variable is $\theta$, so maximizing $A$ means maximizing $\sin\theta$.
Step 3: Find the maximum of sin(theta).
The maximum value of $\sin\theta$ is $1$, which occurs at $\theta = 90^\circ$. So $A_{\max} = \frac{1}{2}ab \cdot 1 = \frac{1}{2}ab$.
Step 4: Confirm using calculus.
Differentiating: $\frac{dA}{d\theta} = \frac{1}{2}ab\cos\theta = 0 \Rightarrow \theta = 90^\circ$. Second derivative: $\frac{d^2A}{d\theta^2} = -\frac{1}{2}ab\sin\theta < 0$ at $\theta = 90^\circ$, confirming a maximum.
Step 5: Interpret geometrically.
When the two given sides are perpendicular, one side acts as the base and the other as the height. This gives the largest possible enclosed area for the given side lengths.
Step 6: State the final answer.
The area of the triangle is maximum when the angle between the two given sides is \[\boxed{90^\circ}\]