Step 1: Identify the known values and the unknown value in the right-angled triangle. We are given: • An angle $\theta = 40^\circ$. • The side adjacent to the $40^\circ$ angle is $8$. • The side opposite to the $40^\circ$ angle is $x$. Step 2: Choose the appropriate trigonometric ratio. The tangent function relates the opposite side, the adjacent side, and the angle: $$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} $$ Step 3: Substitute the given values into the formula. $$ \tan(40^\circ) = \frac{x}{8} $$ Step 4: Solve for $x$. Multiply both sides by $8$: $$ x = 8 \times \tan(40^\circ) $$ Step 5: Calculate the value of $x$. Using a calculator for $\tan(40^\circ)$: $$ \tan(40^\circ) \approx 0.8391 $$ $$ x = 8 \times 0.8391 $$ $$ x \approx 6.7128 $$ Step 6: Round the answer to two decimal places. $$ x \approx 6.71 $$ The value of $x$ is: $$ \boxed{\text{6.71}} $$