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JEE PhysicsJEE Chemistry
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JEE Maths
Trigonometry

Trigonometry 

Trigonometry is that branch of mathematics that relates to the sides and angles of triangles, especially in a right-angled triangle. In solving a vast array of problems, one has to learn the basic principles of trigonometric functions and identities. Below, we discuss the major topics in trigonometry IIT JEE maths and how they help in JEE preparation.

1.0Key Concepts of Trigonometry 

1. Trigonometric Ratios and identities 

In a right-angled triangle, there exist six fundamental trigonometric ratios and identities defining the relationship between the angles and the sides:

 Sine (sin):sinθ= Hypotenuse  Perpendicular ​

 Cosine (cos):cosθ= Hypoternuse  Base ​

 Tangent (tan):tanθ= Base  Perpendicular ​

 Cotangent (cot):cotθ=tanθ1​= Perpendicular  Base ​

 Secant (sec): secθ=cosθ1​= Base  Hypotenuse ​

 Cosecant (csc): cosecθ=sinθ1​= Perpendicular  Hypotenuse ​


0

30

45

60

90

Sine

0

1 / 2 


1/2​



3​/2


1

cos

1


3​/2



1/2​


1 / 2 

0

tan

0


1/3​


1


3​


ND

sec

1


2/3​



2​


2

ND

cosec

ND

2


2​



2/3​


1

cot

ND


3​


1


1/3​


0

2. Trigonometric Identities

Trigonometric identities play a crucial role in simplifying trigonometric expressions and solving equations. Some important identities include:

Pythagorean Identity

sin2θ+cos2θ=1

1+tan2θ=sec2θ

1+cot2θ=cosec2θ

Reciprocal Identities

secθ=cosθ1​⋅cosecθ=sinθ1​,cotθ=tanθ1​

Sum and Difference Identities

sin(A±B)=sinAcosB±cosAsinB

cos(A±B)=cosAcosB∓sinAsinB

tan(A±B)=1∓tanAtanBtanA±tanB​

cot(A±B)=cotB±cotAcotA⋅cotB∓1​

sin(A+B)sin(A−B)=sin2A−sin2B=cos2B−cos2A

cos(A+B)cos(A−B)=cos2A−sin2B=cos2B−sin2A

\sin (A+B+C)=\sinA \cos B \cos C+\sin B \cos A \cos C+\sin C \cos A \cos B-\sin A \sin B \sin C

cos(A+B+C)=cosAcosBcosC−cosAsinBsinC−cosBsinAsinC−cosCsinAsinB

tan(A+B+C)=1−tanAtanB−tanBtanC−tanCtanAtanA+tanB+tanC−tanAtanBtanC​

Transformation Formula

2sinAcosB=sin(A+B)+sin(A−B)

2sinBcosA=sin(A+B)−sin(A−B)

2cosAcosB=cos(A+B)+cos(A−B)

2sinAsinB=cos(A−B)−cos(A+B)

sinA+sinB=2sin(2A+B​)cos(2A−B​)

sinA−sinB=2cos(2A+B​)cos(2A−B​)

cosA+cosB=2cos(2A+B​)cos(2A−B​)

cosA−cosB=2sin(2A+B​)sin(2A−B​)

Multiple Angle Ratios

sin2A=2sin Acos A=1+tan2A2tanA​

cos2A=2cos2A−1=1−2sin2A=1+tan2A1−tan2A​=cos2A−sin2A

tan2A=1−tan2A2tanA​

sin3A=3sinA−4sin3A

cos3A=4cos3A−3cosA

tan3A=1−3tan2A3tanA−tan3A​

3. Trigonometric Functions of Angles

The trigonometric functions are further defined on the unit circle. This defines the concept of sine, cosine, tangent, and other related functions for all angles and just right triangles. The periodicity of the trigonometric circular function is deeper with a unit circle.

Unit Circle Definition: For an angle, , the sine and cosine are defined as:

sinθ=y-coordinate ,   cosθ=x - coordinate, 

and

tanθ=cosθsinθ​

This can be visualized by seeing the behaviour of the values of the trigonometric functions for angles larger than 90°, that is, in the second, third, and fourth quadrants.

Unit Circle Definition

4. Trigonometric Equations

The Trigonometric equations often involve using the basic trigonometric ratios and identities for determining the measures of unknown angles. For instance, general equations for these trigonometric functions are:

Sinθ = Sinα, and the general solution is θ = nπ + (-1)nα, where n ∈ Z

Cosθ = Cosα, and the general solution is θ = 2nπ + α, where n ∈ Z

Tanθ = Tanα, and the general solution is θ = nπ + α, where n ∈ Z

2.0Graphs of Trigonometric Functions

Trigonometric Function 

Graphs

Sine 

y=sinx

Domain is R

Range (-1,1) 

Graph of Sinx


Cos

y=cosx

Domain is R

Range (-1,1)

Graph of Cost


Tan

y=tanx

Domain = R - (2n+1)π/2

Range = R

Graph of Tanx


Cot

y= cotx

Domain = R - nπ

Range = R

Graph of Cotx


Sec

y= secx

Domain: R - {(2x+1)π/2}

Range: (-\infty,-1) \cup(1, \infty)

Graph of Secx


Cosec

y = cosecx

Domain: R - {nπ}

Range : (-\infty,-1) \cup(1, \infty)

Graph of Cosecx



3.0Solved Problems 

Question 1: If

Sinx=−53​, where π<x<23π​, then 80(tan2x−cosx)is equal to

Ans: 109

Explanation

sinx=−53​ where π<x<23π​

tanx=43​,cosx=5−4​

80(tan2x−cosx)

=80(169​+54​)

=(8045+64​)

=109


Question2: Suppose

θ∈[0,4π​] is a solution of 4cosθ−3sinθ=1.  

Then

cosθ is equal to:

Ans:

(36​−2)4​

Explanation

4cosθ−3sinθ=1

4cosθ−1=3sinθ

Squaring Both sides

16cos2θ−8cosθ+1=9(1−cos2θ)

16cos2θ−8cosθ−8=−9cos2θ

⇒25cos2θ−8cosθ−8=0

⇒cosθ=2.258±64+4×25×8​​

=2.258±44+50​​

=254±254​​

 As θ∈[0,4π​]

⇒cosθ=254+66​​=36−2​4​


Question 3: Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2 (α + β) = cos 2α 

Solution: LHS 

= 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) 

= 2 sin2β + 4 (cos α cos β – sin α sin β) sin α sin β + (cos 2α cos 2β – sin 2α sin 2β) 

= 2 sin2β + 4 sin α cos α sin β cos β – 4 sin2α sin2β + cos 2α cos 2β – sin 2α sin 2β 

(4 sin α cos α sin β cos β = 2 sin α cos α .2sin β cos β)

(2 sin α cos α =sin 2α and 2sin β cos β=sin 2β) 

= 2 sin2β + sin 2α sin 2β – 4 sin2α sin2β + cos 2α cos 2β – sin 2α sin 2β 

(cos 2β = 1–2 sin2β)

= (1 – cos 2β) – (2 sin2α) (2 sin2β) + cos 2α cos 2β 

= (1 – cos 2β) – (1 – cos 2α) (1 – cos 2β) + cos 2α cos 2β 

= (1 – cos 2β)(1–1+cos 2α) + cos 2α cos 2β

= (1 – cos 2β)(cos 2α) + cos 2α cos 2β

= cos 2α – cos 2α cos 2β + cos 2α cos 2β

= cos 2α 


Problem 4: Prove that Sec8-1sec4-1=tan8tan2

Solution: LHS

Sec8-1sec4-1=1/cos8-11/cos4-1

=(1-cos8)cos4cos8(1-cos4)

(cos2 = 1 – sin2; sin2 = 1 – cos2; sin24 = 1 – cos8)

=2sin24cos4cos8sin22

=sin4(2sin4cos4)2cos8sin22

(sin2 = 2sincos; sin8 = 2sin4cos4)

=sin4sin82cos8sin22

=2sin2cos2sin8sin22cos8

(sincos=tan)

= tan8tan2

Table of Contents


  • 1.0Key Concepts of Trigonometry 
  • 1.11. Trigonometric Ratios and identities 
  • 1.22. Trigonometric Identities
  • 1.2.1Pythagorean Identity
  • 1.2.2Reciprocal Identities
  • 1.2.3Sum and Difference Identities
  • 1.2.4Transformation Formula
  • 1.2.5Multiple Angle Ratios
  • 1.33. Trigonometric Functions of Angles
  • 1.44. Trigonometric Equations
  • 2.0Graphs of Trigonometric Functions
  • 3.0Solved Problems 

Frequently Asked Questions

Trigonometry is crucial when solving wave, oscillatory, and circular motion problems in physics. It appears in applications of calculus, coordinate geometry, and further mathematics as well.

The basic trigonometric ratios are sine, cosine, tangent, cotangent, secant, and cosecant, which are all derived from a right-angled triangle.

Trigonometric identities are equations that link trigonometric functions together. They reduce complex trigonometric expressions and form the backbone of solving any problem in JEE.

A unit circle is a circle with radius 1, and it defines the trigonometric functions for all angles. Beyond 90°, this is critical for understanding what happens to the functions.

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