What Is Trigonometry? Introduction, Types, Formulas, and Real-Life Applications

Have you ever wondered how engineers measure the height of a mountain without climbing it?

It may sound complicated, but the answer lies in a simple idea from Mathematics.

Trigonometry, a branch of Mathematics, helps us understand the relationship between the sides and angles of a right-angled triangle and use it to solve real-world problems.

It helps us find unknown sides or angles when certain values are already given. At the heart of Trigonometry are three key ratios which compare the lengths of the triangle’s sides.

Angles are usually measured in degrees such as 0°, 30°, 45°, 60°, and 90°. Let us learn more about this topic today:

Introduction to Trigonometry

Before answering what is Trigonometry, it is important to have a basic understanding of its origins.

The term Trigonometry comes from the Greek words ‘trigonon’, meaning triangle, and ‘metron’ meaning measure.

Its roots date back to ancient civilisations, where early ideas related to triangles were used in construction and astronomy.

Trigonometry is a branch of Mathematics that studies the relationships between the sides and angles of a right-angled triangle.

These trigonometric relationships are expressed through trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant.

The term itself became common as a 16th century Latin derivative, and it was conceptualised by Hipparchus, one of the earliest Greek mathematicians.

Applications of Trigonometry

The applications of Trigonometry are varied and extend to different fields of study, such as Physics, Chemistry, and Engineering.

It is not only a theoretical branch of Mathematics but also a practical tool used by professionals on a day-to-day basis.

Trigonometric calculations are frequently used by engineers in construction and design. It is also used by surveyors, architects, and pilots to provide them with the right measurements and direction.

In ancient times, Hipparchus used trigonometric concepts in astronomy. It is still used today to assist astronomers in measuring the distances between stars and determining their locations. It is also needed in the satellite navigation systems.

Trigonometry is particularly employed where measurements are hard to determine. It assists in determining the height of mountains, depth of the lake, or the size of irregular landforms. Trigonometric principles are also used to accurately measure time by astronomers.

In addition to science and measurements, trigonometry is also used in gaming and music. In video games, it is used to regulate the movement of objects. Trigonometric functions are applied in music to examine sound waves and identify pitch.

The uses of trigonometry extend far beyond the classroom. It is a useful and powerful tool in daily life, beginning with space research and all the way up to entertainment.

Types of Trigonometry

Trigonometry is further divided into two sub-branches:

• Plane Trigonometry: Deals with triangles on a flat surface
• Spherical Trigonometry: Deals with triangles on a curved surface

Trigonometric Basics

The Trigonometry basics pertain to measuring angles and solving problems related to them.

The concept is based on three primary trigonometric functions: sine, cosine, and tangent. These core ratios form the foundation for the other three functions, i.e., cotangent, secant, and cosecant.

To understand Trigonometry, it is important that you first learn these basic functions and their formulas. All concepts related to Trigonometry are based on these functions.

Let us take a right-angled triangle, where θ is one of the angles. The three main trigonometric ratios are defined as:

Sin θ = Perpendicular / Hypotenuse
Cos θ = Base / Hypotenuse
Tan θ = Perpendicular / Base

Here, the perpendicular is the side opposite angle θ, the base is the side adjacent to angle θ, and the hypotenuse is the side opposite the right angle (and the longest side of the triangle).

The remaining three trigonometric functions are derived from these primary ratios:

• Cot θ = 1 / Tan θ
• Sec θ = 1 / Cos θ
• Cosec θ = 1 / Sin θ

These can also be expressed as:

• Cot θ = Base / Perpendicular
• Sec θ = Hypotenuse / Base
• Cosec θ = Hypotenuse / Perpendicular

Together, these six functions constitute the basic framework of trigonometry.

Trigonometric Formulas

In Trigonometry, identities are important formulas that help simplify and solve complex problems quickly.

When you use these Trigonometric formulas, you can convert the difficult trigonometric expressions to simpler forms.

Below is the formula of trigonometry, grouped by type.

Reciprocal Identities

These identities show the relationship between a function and its reciprocal:

• Sin θ = 1 / Csc θ or Csc θ = 1 / Sin θ
• Cos θ = 1 / Sec θ or Sec θ = 1 / Cos θ
• Tan θ = 1 / Cot θ or Cot θ = 1 / Tan θ

Pythagorean Identities

These identities are derived from the Pythagoras theorem for right-angled triangles:

• sin² a + cos² a = 1
• 1 + tan² a = sec² a
• cosec² a = 1 + cot² a

Ratio Identities

These express tangent and cotangent in terms of sine and cosine:

• Tan θ = Sin θ / Cos θ
• Cot θ = Cos θ / Sin θ

Identities of Opposite Angles

These show how trigonometric functions behave with negative angles:

• Sin (−θ) = −Sin θ
• Cos (−θ) = Cos θ
• Tan (−θ) = −Tan θ
• Cot (−θ) = −Cot θ
• Sec (−θ) = Sec θ
• Csc (−θ) = −Csc θ

Identities of Complementary Angles

Two angles are complementary if their sum is 90°. Their trigonometric relationships are:

• Sin (90 − θ) = Cos θ
• Cos (90 − θ) = Sin θ
• Tan (90 − θ) = Cot θ
• Cot (90 − θ) = Tan θ
• Sec (90 − θ) = Csc θ
• Csc (90 − θ) = Sec θ

Identities of Supplementary Angles

Two angles are supplementary if their sum is 180°. Their trigonometric identities are:

• sin (180° − θ) = sin θ
• cos (180° − θ) = −cos θ
• cosec (180° − θ) = cosec θ
• sec (180° − θ) = −sec θ
• tan (180° − θ) = −tan θ
• cot (180° − θ) = −cot θ

Sum and Difference of Angles Trigonometric Identities

Consider two angles, α and β, the trigonometric sum and difference identities are as follows:

• sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
• sin(α–β)=sinα.cosβ–cosα.sinβ
• cos(α+β)=cosα.cosβ–sinα.sinβ
• cos(α–β)=cosα.cosβ+sinα.sinβ

Double Angle Trigonometric Identities

If the angles are doubled, then the trigonometric identities for sin, cos and tan are:

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos2θ – sin2 θ = 2 cos2θ – 1 = 1 – 2sin2 θ
• tan 2θ = (2tanθ)/(1 – tan2θ)

Half Angle Identities

If the angles are halved, then the trigonometric identities for sin, cos and tan are:

• sin (θ/2) = ±√[(1 – cosθ)/2]
• cos (θ/2) = ±√(1 + cosθ)/2
• tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

Product-Sum Trigonometric Identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

• Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2
• Cos A + Cos B = 2 Cos(A+B)/2 . Cos(A-B)/2
• Sin A – Sin B = 2 Cos(A+B)/2 . Sin(A-B)/2
• Cos A – Cos B = -2 Sin(A+B)/2 . Sin(A-B)/2

Trigonometric Identities of Products

These identities are:

• Sin A. Sin B = [Cos (A – B) – Cos (A + B)]/2
• Sin A. Cos B = [Sin (A + B) + Sin (A – B)]/2
• Cos A. Cos B = [Cos (A + B) + Cos (A – B)]/2

Trigonometry Table

A trigonometry table contains the values of the main trigonometric ratios, which are all connected to one another. These ratios include sine, cosine, tangent, cosecant, secant, and cotangent.

They are commonly written in short form as sin, cos, tan, cosec, sec, and cot. The table lists their values for standard angles. This makes it easier to solve trigonometric problems quickly.

Key Takeaways

Trigonometry is not just limited to the concept of simple triangles and angle measurements; its impact goes beyond the classroom studies.

Trigonometry is used to solve several real-life problems, such as helping engineers design buildings, astronomers in satellite navigation and more.

It is introduced in middle school and explored in greater depth in senior secondary classes. Triangles are a vast subject ,and Trigonometry is one of its crucial components.

You can also check out our article on Common Mistakes Made by Students in Similarity of Triangles in CBSE Class 10.

FAQs

Q1. What is meant by Trigonometry?

A1: Trigonometry is a branch of Mathematics that studies the relationship between the sides and angles of triangles.

Q2. Is Trigonometry hard?

A2: Trigonometry can seem challenging at first, but it becomes easier with practice and a clear understanding of formulas and concepts. Regular problem-solving builds confidence.

Q3. Why is Trigonometry important?

A3: Trigonometry is important because it helps in solving problems related to angles, distances, and heights. It is widely used in Science, Engineering, and Architecture.

Q4. Where is Trigonometry used?

A4: Trigonometry is used in fields like Engineering, Astronomy, Surveying, Navigation, and Physics. It also plays a role in computer graphics and game development.

Q5. Why do we need Trigonometry?

A5: We need Trigonometry to measure distances and heights that are difficult to calculate directly. It helps in accurate design, construction, and scientific calculations.

Q6. Who is the founder of Trigonometry?

A6: Hipparchus, a Greek astronomer, geographer, and mathematician, is often called the founder of Trigonometry. He developed early trigonometric concepts used in Astronomy.

Q7. How is Trigonometry used in real life?

A7: Trigonometry is used to measure the heights of buildings and mountains, design structures, and navigate using GPS. It is also applied in music, aviation, and satellite systems.