TRIGONOMETRY

TRIGONOMETRY

Trigonometry is one of the most important branches of mathematics that deals with the relationship between the ratios of the sides of a right-angled triangle with its angles. The ratios used to study this relationship are trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. The word trigonometry is a 16th-century Latin derivative and the concept was given by the Greek mathematician Hipparchus. 

INTRODUCTION TO TRIGONOMETRY

The word trigonometry is formed by clubbing words ‘Trigonon’ and ‘Metron’ which mean triangle and measure respectively. It is the study of the relation between the sides and angles of the right-angled triangle. It thus helps in finding the measure of unknown dimensions of the right-angled triangle using formulas and identities based on this relationship.

TRIGONOMETRY BASICS

Trigonometry basics deal with the measurement of angles and problems related to angles. These are three basic functions in trigonometry: sine, cosine and tangent. These three basic ratios or functions can be used to derive the important trigonometric functions: cotangent, secant and cosecant. All the important concepts covered under trigonometry are based on these functions. In a right-angled triangle, there are the following three sides.

• Perpendicular - It is the side opposite to the angle θ.

• Base - This is the adjacent side to the angle θ.

• Hypotenuse - This is the side opposite to the right angle.

TRIGONOMETRIC RATIOS

There are basic six ratios in trigonometry that help in establishing a relationship between the ratio of sides of a right triangle with the angle. If θ is the angle in a right-angled triangle, formed between the base and the hypotenuse, then

• sin θ = Perpendicular/Hypotenuse

• cos θ = Base/Hypotenuse

• tan θ = Perpendicular/Base

The value of the other three functions: cot, sec and cosec depend on tan, cos, and sin respectively as given below.

• cot θ = 1/tan θ = Base/Perpendicualr

• sec θ = 1/cos θ = Hypotenuse/Base

• cosec θ = 1/sin θ = Hypotenuse/Perpendicular

TRIGONOMETRIC TABLE

The trigonometric table is made up of trigonometric ratios that are interrelated to each other sine, cosine, tangent, cosecant, secant and cotangent. These ratios, in short, are written as sin, cos, tan, cosec, sec, and cot and are taken for standard angle values. One can refer to the trigonometric chart to know more about these ratios. 

IMPORTANT TRIGNOMETRIC ANGLES

Trigonometric angles are the angles in a right-angled triangle using which different trigonometric functions can be represented. Some standard angles used in trigonometry are 0º, 30º, 45º, 60º and 90º. The trigonometric values for these angles can be observed directly in a trigonometric table. Some other important angles in trigonometry are 180º, 270º, and 360º. Trigonometry angle can be expressed in terms of trigonometric ratios as,

• θ = sin-1 (Perpendicular/Hypotenuse)

• θ = cos-1 (Base/Hypotenuse)

• θ = tan-1 (Perpendicular/Base)

LIST OF TRIGONOMETRIC FUNCTIONS

Different formulas in trigonometry depict the relationships between trigonometric ratios and the angles for different quadrants. The basic trigonometry formulas list is given below:

1. Trigonometry Ratio Formulas

• sin θ = Opposite Side/Hypotenuse

• cos θ = Adjacent Side/Hypotenuse

• tan θ = Opposite Side/Adjacent Side

• cot θ = 1/tan θ = Adjacent Side/Opposite Side

• sec θ = 1/cos θ = Hypotenuse/Adjacent Side

• cosec θ = 1/sin θ = Hypotenuse/Opposite Side

2. Trigonometry Formulas Involving Pythagorean Identities

• sin²θ + cos²θ = 1

• tan2θ + 1 = sec2θ

• cot2θ + 1 = cosec2θ

3. Sine and Cosine Law in Trigonometry

• a/sinA = b/sinB = c/sinC

• c2 = a2 + b2 – 2ab cos C

• a2 = b2 + c2 – 2bc cos A

• b2 = a2 + c2 – 2ac cos B

Here a, b, and c are the lengths of the sides of the triangle and A, B, and C are the angles of the triangle.

TRIGONOMETRIC FUNCTIONS GRAPHS

Different properties of a trigonometric function like domain, range, etc. can be studied using the trigonometric function graphs. The graphs of basic trigonometric functions- Sine and Cosine are given below:

The domain and range of sin and cosine functions can thus be given as,

• sin θ: Domain (-∞, + ∞); Range [-1, +1]

• cos θ: Domain (-∞ +∞); Range [-1, +1]

UNIT CIRCLE AND TRIGONOMETRIC VALUES

Unit circles can be used to calculate the values of basic trigonometric functions- sin, cosine, and tangent. The following diagram shows how trigonometric ratios sine and cosine can be represented in a unit circle.

TRIGONOMETRIC IDENTITIES

In Trigonometric Identities, an equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved. In trigonometric identities, we will get to learn more about the Sum and Difference Identities.

For example, 

sin θ/cos θ = [Opposite/Hypotenuse] ÷ [Adjacent/Hypotenuse]

= Opposite/Adjacent

 = tan θ

Therefore, tan θ = sin θ/cos θ is a trigonometric identity. 

APPLICATIONS OF TRIGONOMETRY

Throughout history, trigonometry has been applied in areas such as architecture, celestial mechanics, surveying, etc. Its applications include in:

• Various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, and many more.

• It is also helpful to find the distance of long rivers, measure the height of the mountains, etc.

• Spherical trigonometry has been used for locating solar, lunar, and stellar positions.

REAL-LIFE EXAMPLE OF TRIGONOMETRY

Trigonometry has many real-life examples used broadly. For example,

A boy is standing near a tree. He looks up at the tree and wonders “How tall is

the tree?” The height of the tree can be found without actually measuring it. What

we have here is a right-angled triangle, i.e., a triangle with one of the angles

equal to 90 degrees. Trigonometric formulas can be applied to calculate the

height of the tree if the distance between the tree and the boy, and the angle

formed when the tree is viewed from the ground is given.

It is determined using the tangent function, such as the tan of angle is equal to the ratio of the height of the tree and the distance. Let us say the angle is θ, then

tan θ = Height/Distance between object & tree
Distance = Height/tan θ

Let us assume that the distance is 30m and the angle formed is 45 degrees, then

Height = 30/tan 45°
Since, tan 45° = 1
So, Height = 30 m

The height of the tree can be found out by using basic trigonometry formulas.

IMPORTANT NOTES ON TRIGONOMETRY

• Trigonometric values are based on the three major trigonometric ratios: Sine, Cosine, and Tangent.
  Sine or sin θ = Side opposite to θ / Hypotenuse
  Cosine or cos θ = Adjacent side to θ / Hypotenuse
  Tangent or tan θ = Side opposite to θ / Adjacent side to θ

• 0°, 30°, 45°, 60°, and 90° are called the standard angles in trigonometry.

• The trigonometry ratios of cosθ, secθ are even functions, since cos(-θ) = cosθ, sec(-θ) = secθ.