Trigonometrical Ratios

The study of the three angles, three sides, and their mutual relationships in a triangle is an important part of trigonometry. To understand these relationships clearly, trigonometrical ratios are defined.

In this lesson, we will learn the six trigonometrical ratios of a positive acute angle and their mutual relationships in a simple way.

Concept of Trigonometrical Ratios

🔹 Trigonometrical Ratios

Suppose OB is a rotating raya ray that can rotate around its initial point while the initial point remains fixed.. It has rotated anticlockwise about the point O from the position OA to the position OB. As a result, ∠AOB is formed.

Let ∠AOB = θ, where θ is an acute angle.

Now take any point P on the ray OB. From P, draw PM perpendicular to OA. Then △POM is a right-angled triangle.

In △POM, OP is called the hypotenuse, PM, the side opposite to the angle θ, is called the perpendicular, and OM, the side adjacent to the angle θ, is called the base.

In short:

Hypotenuse = OP = h

Perpendicular = PM = p

Base = OM = b

Introduction to the Ratios

In a right-angled triangle, with respect to the angle θ, the ratio of the perpendicular to the hypotenuse is called sin θ.

Therefore,

∴ sin θ =

Similarly, the other five ratios can be explained.

The six trigonometrical ratios are:

💡 Easy Way to Remember

A simple and common way to remember the first three ratios is:

After that, the remaining three ratios are easy to find because they are the reciprocal ratios of the first three ratios.

Reciprocal Relations

📚 Important Points

👉 sin θ, cos θ, tan θ, and the other trigonometrical ratios are ratios of two lengths. So, they have no unit.

👉 sin θ does not mean sin × θ. It means the ratio of the perpendicular and the hypotenuse with respect to the angle θ.

👉 The side opposite to the right angle is called the hypotenuse. The side opposite to the angle θ is called the perpendicular, and the side adjacent to the angle θ is called the base.

👉 The trigonometrical ratio of a fixed angle depends only on the value of that angle. Even if the size of the triangle changes, the ratio remains the same as long as the angle remains unchanged.

👉 In trigonometry, writing only sin, cos or tan is incomplete. They should be written with a particular angle, such as sin θ, cos θ, tan θ.

👉 sin² θ means (sin θ)². But sin θ² means sin(θ²). So, sin² θ and sin θ² are not the same.

👉 Similarly, (cos θ)⁵ = cos⁵ θ and (tan θ)⁷ = tan⁷ θ can be written.

From the Formula of a Right-Angled Triangle

We know that in a right-angled triangle, the sum of the square of the perpendicular and the square of the base is equal to the square of the hypotenuse.

That is, p² + b² = h²

Dividing both sides by h², we get:

Again, p² + b² = h²

Dividing both sides by b², we get:

And, p² + b² = h²

Dividing both sides by p², we get:

📚 Quick Revision

✅ Hypotenuse = h, perpendicular = p, and base = b