Trigonometry

Introduction to Trigonometry

Trigonometry is defined as one of the branches of mathematics that deals with the relationships that involve lengths and also the angles of triangles. In a simple manner, we can say that Trigonometry is the study of triangles. The word trigonometry was derived from the Greek word, where, ‘TRI’ means Three, ‘GON’ means sides and whereas the ‘METRON’ ways to measure. This concept is given to us by a Greek mathematician Hipparchus.

To be more specific, trigonometry is all about a right-angled triangle, where one of the internal angles measure about 90°. Moreover, it is one of those divisions in mathematics that helps in finding the angles and missing sides of a triangle. In trigonometry, the angles are either measured in radians or degrees.

This branch divides into two sub-branches called plane trigonometry and the spherical geometry. Trigonometry, in general, is about the trigonometric formulas,  trigonometric ratios, and functions, Right-Angled Triangles, etc. Let us study all these topics in detail.

Trigonometric Functions and Ratios:

The trigonometric ratios of a triangle are also called the trigonometric functions. There are three essential trigonometric functions in trigonometry known as Sine, cosine, and tangent and abbreviated as sin, cos, and tan. How are these ratios or functions, evaluated in the case of a right-angled triangle?

Consider the right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse is referred to as the adjacent and opposite. The trigonometric functions or the ratios of this triangle is calculated by the below formulas.

• Sine ratio or Function is given as, sin θ = Opposite / Hypotenuse
• Tangent ratio or Function is given as, tanθ = Opposite / Adjacent
• Cosine ratio or Function is given as, cos θ = Adjacent / Hypotenuse

Similar to the ratios sine, cosine and tangent, there are other three trigonometric ratios or functions in trigonometry called Cotangent,  Cosecant, and Secant. The values of these trigonometric functions are evaluated by using the following formulas

• Cosecant Function (cosec θ) = Hypotenuse / Opposite
• Cotangent Function (cot θ) = Adjacent / Opposite
• Secant Function (sec θ) = Hypotenuse / Adjacent

Inverse Trigonometric Ratios:

The inverse trigonometric functions are those functions which involve the inverses of cosine, tangent, and sine. These opposites, are called inverse trigonometric functions. By considering the right-angled triangle, the inverse functions, are given below-

• cosec θ = 1/sin θ = Hypotenuse/opposite
• cot θ = 1/tan θ = adjacent / opposite
• sec θ = 1/cos θ = hypotenuse/ adjacent

Trigonometric Ratios of Specific Angles:

Suppose if you are given the question – In a right-angled triangle ABC, if one side of the triangle is 45°, then what is the value of other side or angle?
For such questions, the below table will help you out in finding the trigonometric and inverse trigonometric ratios of different Angles of a triangle.

Pythagoras Theorem:

The Pythagoras theorem helps to know the relationship between the trigonometric identities, which, is discussed in the next sub-heading.

Pythagoras theorem states “ the square of the hypotenuse (c )equals the sum of the squares of the adjacent( b ) and opposite ( a).”

In equation form, it is, given as: c=a2 + b2

Trigonometric Identities:

An identity is a form of equation true for all the values of the variables, which both the sides of an equation is defined. These variables are either in the shape of a statement or even specified. The Pythagoras theorem is also one of the trigonometric identities. The Trigonometric Identities are the equations which are true in the case of Right Angled Triangles.

Some of the other identities or rather say formulas in trigonometry, are as given below –

1. Pythagorean Identities-
• sin ² θ + cos ² θ = 1
• tan 2 θ + 1 = sec2 θ
• cot2 θ + 1 = cosec2 θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos² θ – sin² θ
• tan 2θ = 2 tan θ / (1 – tan² θ)
• cot 2θ = (cot² θ – 1) / 2 cot θ
1. Sum and Difference identities-

$$\sin (\pi /2-u)= \cos u \cos (\pi /2-u)= \sin u \tan (\pi /2-u)= \cot u$$

$$cosec(\pi /2-u)= sec(u) sec(\pi /2-u)= cosec (u) cot(\pi /2-u)= tan(u)$$

• Sine Laws-

$$\frac{a}{\sin A}=\frac{b}{\sin B} = \frac{c}{\sin C} = 2R = \frac{abc}{2 \Delta}$$

$$Area = \Delta = \frac{1}{2}ab \sin C$$

• Cosine Laws-

$$c^{2} = a^{2} + b^{2} – 2ab \cos C$$

• Tangent Laws-

$$\frac{a-b}{a+b} = \frac{\tan \left ( \frac{\alpha – \beta}{2} \right )}{\tan \left ( \frac{\alpha + \beta}{2} \right )}$$

Applications of Trigonometry

Trigonometry finds its applications in various fields like seismology,  oceanography, meteorology, physical sciences, navigation, astronomy, acoustics, electronics, etc.

It is also helpful to measure the heights of the mountain, find the distance of long rivers, etc.

Learn about Trigonometry in a simple manner with detailed information along with step by step solutions to all questions of this chapter in an NCERT Books only at BYJU’S.