Trigonometric ratios are essential tools for finding a missing side in a right-angled triangle. To apply these ratios effectively, you must first identify the sides in relation to the given angle. The three sides of a right triangle are defined as follows:

• Opposite: The side opposite the angle in question.
• Adjacent: The side that is next to the angle.
• Hypotenuse: The longest side, which is always opposite the right angle.

The three primary trigonometric ratios are:

1. Sine (sin): $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
2. Cosine (cos): $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
3. Tangent (tan): $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

To determine the appropriate ratio to use, consider the sides you know and the side you need to find. For example, if you have the length of the hypotenuse and the measure of the angle, but need to find the length of the opposite side, you would use the sine function.

Here is a step-by-step guide to finding a missing side:

1. Identify the given angle and the sides involved.
2. Select the appropriate trigonometric ratio based on the known sides.
3. Substitute the known values into the chosen ratio.
4. Rearrange the equation to isolate the missing side.
5. Utilize a calculator to compute the value.

Example:

Suppose you know that the angle is $30^\circ$ and the length of the hypotenuse is $10$ units, and you need to find the length of the opposite side. You would apply the sine function as follows:

$\sin(30^\circ) = \frac{\text{Opposite}}{10}$

Knowing that $\sin(30^\circ) = 0.5$, we can substitute this into the equation:

$0.5 = \frac{\text{Opposite}}{10}$

To solve for the opposite side, multiply both sides by $10$:

$\text{Opposite} = 0.5 \times 10$

This gives:

$\text{Opposite} = 5 \text{ units}$

By following these steps, you can effectively utilize trigonometric ratios to find any missing side in a right-angled triangle.