# Introduction to Trigonometry: Ratios and Applications in Geometry

*In this article, we’ll explore the basics of triangles, trigonometric ratios, and how they’re used in geometry, with a particular focus on the right-angle triangle. Join us on this journey to unravel the fascinating world of trigonometry and its practical applications in geometry!*

## The Right Triangle

All the trigonometric ratios that will be discussed in this article describe the intrinsic properties of right triangles specifically. A right triangle is a triangle that has two legs that are perpendicular to each other, i.e. one of its interior angles is a right angle.

When studying trigonometric ratios in right triangles, the following terms are used to describe various elements of a right triangle:

**Hypotenuse**: The longest side of a right triangle, directly opposite the right angle.**Opposite**: The side that is directly across from the chosen angle (θ on the diagram) within the right triangle.**Adjacent**: The side that, together with the hypotenuse, forms the chosen angle (θ on the diagram) within the right triangle.

## Trigonometric Ratios

In right triangles, the relationship between pairs of sides remains constant and is not affected by the size of the triangle, provided the angle of reference remains identical. These consistent relationships between the sides are known as trigonometric ratios.

The three main trigonometric ratios are Sine (sin), Cosine (cos), and Tangent (tan).

## The Sine Ratio

Let’s take the image above as an example. The sine ratio corresponding to a reference angle is the definition given to the ratio between the opposite side and the hypotenuse of a right triangle. In simpler terms, if you want to find the sine value of the angle labeled α it would be as follows:

## The Cosine Ratio

The cosine ratio associated with a reference angle is defined as the ratio between the length of the adjacent side and the length of the hypotenuse in a right triangle. In simpler terms, to determine the cosine value of the angle labeled α, you would compute it as follows:

## The Tan Ratio

The tangent ratio corresponding to a reference angle is defined as the ratio between the length of the side opposite to the angle and the length of the side adjacent to the angle in a right triangle. To find the tangent value of the angle labeled α, you would calculate it as follows:

Trigonometric ratios in geometry serve the purpose of finding missing sides in right angles. If at least one angle and the length of a side are known, the length of the remaining sides can be determined using an appropriate trigonometric ratio.

**Example: Determine the value of x**

Since the opposite side is known, and the hypotenuse is to be calculated, we must use a ratio that includes those two sides. This is the Sin Ratio (Opposite/ Hypotenuse).

Step 3: Apply the ratio and simplify

## Applications of trigonometry in Geometry

In architecture, trigonometry helps architects and engineers design buildings with precision. Imagine you’re standing far away from a tall building and want to know its height. By measuring the angle formed between the ground and your line of sight to the top of the building, you can use trigonometric principles to calculate its height.

Moreover, in everyday situations, such as navigating through a city or hiking on a trail, trigonometry comes in handy for estimating distances and heights. By using landmarks and measuring angles, individuals can determine their position and plan their routes effectively.

Whether it’s finding the height of a tree or the distance to a destination, trigonometric ratios provide practical solutions to real-world problems in geometry.

## Conclusion

As you can see trigonometry is a powerful tool that allows us to solve complex geometric problems and explore the world around us. By understanding trigonometric ratios, you can explore the relationships between angles and sides of triangles.