Trigonometric ratios are super helpful when we need to find heights and distances, especially when we can’t measure them directly. Let’s make it easier to understand!

What Are Trigonometric Ratios?

In right-angled triangles, we mostly use three important ratios: sine, cosine, and tangent. These help us connect the triangle's angles with the lengths of its sides.

1. Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
   (Opposite side over Hypotenuse)

2. Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
   (Adjacent side over Hypotenuse)

3. Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
   (Opposite side over Adjacent side)

Finding Heights with Trigonometric Ratios

Let’s say you want to figure out how tall a tree is. You stand 20 meters away and look up at the tree. If you measure the angle from where you are to the top of the tree and it’s $30^\circ$, you can find the height!

You can use the tangent ratio for this:
$h = d \cdot \tan(\theta)$
Here, $h$ is the height and $d$ is how far you are from the tree.
So it looks like this:
$h = 20 \cdot \tan(30^\circ) \approx 20 \cdot 0.577 \approx 11.54 \, \text{meters}$
That means the tree is about 11.54 meters tall!

Finding Distances with Trigonometric Ratios

Now, let’s say you want to find out how far away you are from a building. You see the top of the building at a $45^\circ$ angle from where you are standing, and you’re 10 meters back from it. You can use the tangent again here:
$d = h \cdot \frac{1}{\tan(45^\circ)} = h$
(That’s because $\tan(45^\circ) = 1$!)

Trigonometric ratios make it easier to find heights and distances. They are really important tools in geometry!