To determine the length of the opposite side in a right-angled triangle, you can utilize either the sine or tangent ratio, depending on the known values—specifically, the angle and either the hypotenuse or the adjacent side.

In trigonometry, the opposite side can be calculated using the sine function ($\sin$) or the tangent function ($\tan$), based on the information you have.

1. Using the Sine Ratio: If you know the length of the hypotenuse and an angle (other than the right angle), you can apply the sine ratio. The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. This relationship can be expressed mathematically as:

   $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

   To isolate the opposite side, you can rearrange the equation:

   $$\text{opposite} = \sin(\theta) \times \text{hypotenuse}$$

   For example, if the hypotenuse measures $10$ units and the angle is $30^\circ$, the calculation for the opposite side would be:

   $$\text{opposite} = \sin(30^\circ) \times 10 = 0.5 \times 10 = 5 \text{ units}$$

2. Using the Tangent Ratio: Alternatively, if you know the length of the adjacent side along with an angle, you can use the tangent ratio. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side, which can be represented as:

   $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

   To find the opposite side, rearrange the equation:

   $$\text{opposite} = \tan(\theta) \times \text{adjacent}$$

   For instance, if the adjacent side measures $8$ units and the angle is $45^\circ$, the calculation for the opposite side would be:

   $$\text{opposite} = \tan(45^\circ) \times 8 = 1 \times 8 = 8 \text{ units}$$

By employing these trigonometric ratios, you can effectively determine the length of the opposite side in a right-angled triangle based on the available measurements.