An Introduction to Trigonometry formulas

Introduction

Trigonometry is the branch of mathematics that deals with the study of triangles to find the distance, height, length of sides and angle of inclination of triangles. In ancient times, astronomers used these trigonometry formulas to estimate the distance between the Earth and other celestial bodies like the Sun, the Moon and stars. Furthermore, Egyptians used theorems that involved ratios of sides of triangles for construction purposes. 

trigonometry formulas
January 12, 2022
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Table of Contents

Trigonometry and Trigonometry Formulas class 10: Overview 

Trigonometry comes from the Greek word ‘trigonon’ that means triangle and ‘metron’ meaning measure. It helps to study the relationship between the length and angles of a triangle. So, before we learn more about trigonometry formulas and the concept of trigonometry in detail. Let us first understand and brush up on some concepts about triangles.  

A triangle is a closed two-dimensional shape that consists of three sides and three corners in geometry. Here are a few key points about a triangle: 

  • It consists of three sides, three vertices and three angles. 
  • The sum of all the three interior angles always equals 180°. 
  • The perimeter of a triangle is the sum of the three sides. 
  • Besides, the area of a triangle is equal to half of the product of its base and height. 

We can classify triangles into three types (based on angles): acute-angled triangle, right-angled triangle, and obtuse-angled triangle. However, this article will only look at the right-angled triangle in detail because trigonometry formulas and concepts are only applicable in right-angled triangles.

So, a right-angled triangle has one right angle equal to 90°. Therefore, it is known as a right-angled triangle. The sum of the other two angles equals 90°. In addition, the sides that form the right angle are the perpendicular and the base. The third side is the longest and is called the hypotenuse. 

The Pythagoras theorem explains the relationship between all the sides of a triangle. According to this theorem, the square of the hypotenuse is equal to the sum of squares of the perpendicular and the base.

Hypotenuse^2 = Perpendicular^2 + Base^2 

Trigonometric ratios 

After learning the basics of the right-angled triangle, let us look at its role in trigonometry formulas and ratios. Consider a triangle ∆ABC which is right-angled at vertices B. So, the trigonometric ratios of angle A are as below: 

  • sin A = opposite side/hypotenuse = BC/AC 
  • cos A = adjacent side/hypotenuse = AB/AC 
  • tan A = opposite side/adjacent side = BC/AB 
  • cosec A = hypotenuse/opposite side = AC/BC 
  • sec A = hypotenuse/adjacent side = AC/AB 

  • cot A = adjacent side/opposite side = AB/BC 

Trigonometry formulas: Relation between trigonometric ratios 

  • cosec θ = 1/sin θ 
  • sin θ = 1/cosec θ 
  • tan θ = sin θ/ cos θ 

  • cot θ = cos θ/ sin θ= 1/tan θ 

As you can see, the reciprocal sin θ is cosec θ; the reciprocal of cos θ is sec θ and the reciprocal of tan θ is cot θ. 

Trigonometric ratios of complementary angles 

If θ is an acute angle then its complementary angle is 90°- θ.  

  • sin ( 90°- θ) = cos θ 

  • cos ( 90° – θ) = sin θ 
  • tan (90° – θ) = cot θ 
  • cot (90° – θ ) = tan θ 
  • cosec (90° – θ) = sec θ 
  • sec (90° – θ) = cosec θ 

Trigonometry Formulas: Trigonometric identities 

We already know the concepts for the Pythagoras theorem, it can also be used to prove the following trigonometric identities:  

  • sin^2 θ + cos^2 θ = 1 
  • 1 + cot^2 θ = cosec^2 θ 
  • 1 + tan^2 θ = sec^2 θ 

Trigonometry Formulas for class 11: Overview 

The basics of trigonometry remain the same for class 10 and beyond. However, as you climb up the stairs your vision grows. You will learn about trigonometry formulas and more of their concepts.  

So, let us recap some basics trigonometry formulas from class 10 

  • sin θ = perpendicular/base 
  • cos θ = base/hypotenuse 

  • tan θ = perpendicular/base 
  • cot θ = base/perpendicular 
  • sec θ = hypotenuse/base 
  • cosec θ = hypotenuse/perpendicular 

Trigonometry functions 

This is a new concept that you may learn. Trigonometry functions are measured in radian for a circle drawn in XY-plane. In short, radian is a measure of an angle, just like a degree. Let us look at the two terms, i.e., degree and radian, in a bit of detail.

Degree: It is a unit of measure for angles. A full circle is divided into 360 degrees. For instance, a right angle is 90 degrees.  

Radian: If an angle is subtended at the centre by an arc of length ‘l’ then the angle is measured as radian. Suppose θ is the angle formed at the centre, then, 

θ = length of the arc/radius of the circle 

θ = l/r 

Relationship between degree and radian 

2π radian = 360° 

Hence, π radian = 180° 

Where, π = 22/7 

Signs for trigonometric functions 

  • sin (-θ) = -sin θ 
  • cos (-θ) = cos θ 
  • tan (-θ) = -tan θ 
  • cot (-θ) = -cot θ 
  • sec (-θ) = sec θ 

  • cosec (-θ) = -cosec θ 

Trigonometry Formulas: Trigonometric functions of Sum and Product of two angles 

1. 

  • sin (x + y) = sin x cos y + cos x sin y 
  • sin (x – y) = sin x cos y – cos x sin y 

  • cos (x + y) = cos x cos y – sin x sin y 
  • cos (x – y) = cos x cos y + sin x sin y 
  • sin (π/2 – x)= cos x 
  • cos (π/2 – x)= sin x 

2. 

  • tan (x+y) = (tan x + tan y) /(1−tan x tan y) 
  • tan (x-y) = (tan x − tan y)/(1 + tan x tan y) 
  • cot (x+y) = (cot x cot y −1)/(cot y + cot x) 
  • cot(x-y) = (cot x cot y + 1)/( cot y − cot x) 

3. 

  • cos 2x = cos2 x – sin2 x = 2cos2 x – 1 = 1 – 2sin2 x = 1 – tan2 x/1 + tan2 x 
  • sin 2x = 2sin x cos x= 2tan x/ 1+ tan2 x 
  • tan 2x=2 tan x/1-tan2 x 

4. 

  • sin3x = 3 sin x – 4 sin3 x 

  • cos 3x = 4 cos3 x – 3 cos x 
  • tan 3x = [3tan x – tan3 x]/[1 – 3 tan2 x] 

5. 

  • cos x + cos y = 2 cos [(x+y)/2] cos [(x−y)/2] 
  • cos x – cos y = −2 sin [(x+y)/2] sin [(x−y)/2] 

  • sin x + sin y = 2 sin [(x+y)/2] cos [(x−y)/2] 
  • sin x – sin y = 2 cos [(x+y)/2] sin [(x−y)/2] 

6. 

  • 2 cos x cos y = cos (x + y) + cos (x – y) 
  • 2 sin x sin y = cos (x – y) – cos (x + y) 

  • and 2 sin x cos y= sin (x + y) + sin (x – y) 
  • finally, 2 cos x sin y = sin (x + y) – sin (x – y) 

B.Sc Mathematics 

Overall, mathematics as a subject is a treasure for students and a nightmare for another group of students. However, for those who like to study mathematics, there is more to explore. While the ones who dislike has a lot to learn. Mathematics is everywhere. We do not comprehend how important and valuable it is on a day to day basis.

B.Sc Mathematics is an honourable degree. Also, the demand for the subject is high all around the globe, especially in India. Not only does the field grant colossal respect, but it also offers flexible career opportunities. It is a three-year undergraduate course dealing with an in-depth study of the subject. The syllabus includes calculus, probability, statistics, linear programming and ring theory.

Eligibility for B.Sc Mathematics 

The eligibility criteria for B.Sc Mathematics is similar to other graduation courses. 

  • Applicants must have passed 10+2 with mathematics as a subject. 
  • The qualifying marks for admission vary from college to college. 

Syllabus for B.Sc Mathematics 

First Year 

  • Calculus 
  • Algebra 
  • Real analysis 
  • Differential equations 

Second Year 

  • Theory of real functions 
  • Group Theory part 1 
  • Multivariate Calculus 
  • Ring Theory 

  • Linear algebra part 1 
  • Riemann integration 

Third Year 

  • Group theory part 2 
  • Complex analysis 

  • Ring theory 
  • Linear algebra part 2 
  • Linear programming 
  • Probability 
  • Statistics 

  • Application of algebra 

Career opportunities after B.Sc. Mathematics 

B.Sc. Mathematics has a lot to offer in career options, from accounting to coding to teaching. We have mentioned some of them below: 

Banking: It is the most common career option after graduation in mathematics. The banking sector in India is growing rapidly, covering around 70% of the market share. Jobs available in the banking sector include Market Researcher, Financial Model Solution Developer and recruitment in some top banking companies like ICICI bank. 

Accounting: Jobs in the accounting sector are increasing, and they grew by 30% in 2019. However, there is a common misconception that only graduates from commerce backgrounds can apply for accounting jobs. But graduates holding a degree in mathematics can also apply. The job profiles in accounting include Auditor Management Accountant, Tax Accountant and Corporate Advisor. 

Government job: Securing a stable and productive job is the ultimate goal of almost all Indians. So, you can also prepare for government exams like SSC CGL, IBPS PO or other civil service examinations to land a successful job in the government sector. 

Teaching: You can also become a teacher by profession after completing B.Sc. in Mathematics. If you can play and reflect the concepts of the subject interestingly, then you can explore your career goals as a teacher. Therefore, you can work as a teacher in school, college or university, or start your coaching centre. 

You can also read our blog on What to do after B.Sc. Mathematics.

Conclusion 

In the above article, we learned the basics of trigonometry formulas and concepts. In addition, we also discussed an overview of B.Sc. Mathematics eligibility, syllabus and career options. Mathematics has a lot to explore, and learning never ends for these subjects. Students must bear patience and must have the ability to study for long hours while choosing such a subject. 

Chegg is one of the leading e-learning platforms in India. They offer work from home opportunities for Subject matter Experts. They are tutors who solve the doubts of students outside India. Suppose you are a B.Sc. Mathematics graduate or hold a minimum of a bachelor’s degree or a diploma in any subject and you have good knowledge about that subject. In that case, you can easily become a Subject Matter Expert at Chegg India. You can choose your choice of subject from a wide range of options like Computer Science, Biology, Physics, Mathematics, Economics and more. Follow these steps to become an SME in Chegg India. 

  • Visit official website of Chegg India. 
  • Register with your email address and sign in. 
  • Secondly, pick a subject of your choice from the subjects available. 
  • Then, you have to appear for an online screening test which is a quick test to check the knowledge of your subject. 
  • If you qualify for the test, you have to verify your documents. 

  • Verification of documents may take 2-3 working days. 

That’s it! Now you can solve questions and start earning with us. The earning potential for a Subject Matter Expert is INR 30,000 to 80,000 per month. 

You can also become a subject matter expert with Chegg.

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