In Year 9 Mathematics, it's really important to understand the basic trigonometric ratios when we look at right triangles.

There are three main ratios you should know:

1. Sine (sin): This compares the length of the side opposite the angle to the longest side, called the hypotenuse.

   • For example, in triangle ABC, if angle A is 30°, then we can say:
     $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$.

2. Cosine (cos): This one compares the length of the side next to the angle (the adjacent side) to the hypotenuse.

   • So, for angle A at 30°, we have:
     $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$.

3. Tangent (tan): This ratio looks at how the opposite side compares to the adjacent side.

   • For angle A at 30°, it is:
     $\tan A = \frac{\text{opposite}}{\text{adjacent}}$.

Let’s make this clearer with an example.

Imagine a right triangle where angle A is 30° and the hypotenuse is 10 units long. We want to find out how long the opposite and adjacent sides are.

Using the trigonometric ratios:

• To find the opposite side:
  $\text{opposite} = 10 \cdot \sin(30°) = 10 \cdot 0.5 = 5 \text{ units}$.

• To find the adjacent side:
  $\text{adjacent} = 10 \cdot \cos(30°) \approx 10 \cdot 0.866 = 8.66 \text{ units}$.

These ratios help us solve many real-life problems with right triangles. They are really important for geometry and can be used in lots of ways to measure things!