How can trigonometric ratios like sine, cosine, and tangent be used to find missing sides or angles in right-angled triangles?

Trigonometric ratios can be used to find missing sides or angles in right-angled triangles by following these steps:

1. Identify the sides: In a right-angled triangle, the side opposite the right angle is the hypotenuse. The side opposite the angle you are interested in is the opposite side, and the remaining side is the adjacent side.

2. Choose the correct ratio: Depending on what you know and what you want to find, choose the correct trigonometric ratio.

   • Sine (sin) is the ratio of the length of the opposite side to the length of the hypotenuse.
   • Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse.
   • Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side.

3. Set up the equation: Write down the trigonometric equation that corresponds to the ratio you have chosen. For example, if you are using sine and you know the angle (let's call it A) and you want to find the length of the opposite side (let's call it o) and you know the length of the hypotenuse (let's call it h), your equation would be sin(A) = o/h.

4. Solve the equation: If you are finding a side, multiply both sides of the equation by the known length to find the unknown length. If you are finding an angle, use the inverse trigonometric function (e.g., sin^-1, cos^-1, tan^-1) to find the angle.

5. Check your answer: Make sure your answer makes sense in the context of the problem. For example, a side length should be positive and an angle should be between 0 and 90 degrees.