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Trigonometric Equations

Trigonometric Equations is a vital component of the IIT JEE Mathematics syllabus. The topic is quite vast and is segregated into various sub topics. Some of the subheads fetch some direct questions which are scoring as well as easy to handle.

In the following sections, we have discussed each sub head in detail. A chief portion of IIT trigonometry questions are asked from Trigonometric equation. As it is a scoring portion, it becomes a deciding factor of your fate in the IIT JEE.

The chapter covers Inverse Circular functions along with their applications to problems including those of trigonometric inequality and extreme values of trigonometric functions.

We fist discuss some of the basic and important concepts which form the groundwork of trigonometry equations:

An equation involving trigonometric functions is called a trigonometric equation.

E.g. Solve sin (x) + 2 = 3 for 0° < x < 360°

Domain: The set of all possible values a function can assume is called the domain of a function.

E.g. Domain of sin x is the set of all Real Numbers. It is so because there is no value of x for which sin x is not defined.

Range: The set of all possible values a function can produce or give is called range.

E.g. Range of sin(x) is all real numbers between -1 and 1. It is clear from the graph of sin x also that sin x is a continuous wave that bounces between -1 and 1. 

The domain and range of trigonometric functions are tabulated below:

Trigonometric Function



sin x

R, the set of real numbers

-1≤ sin x ≤1

cos x

R, the set of real numbers

-1 ≤ cos x ≤1

tan x

R-{(2n+1)π/2, n I}


cosec x

R-{(nπ, n I}

R-{x: -1<x<1}

sec x

R-{(2n+1)π/2, n I}

R-{x: -1<x<1}

cot x

R-{(nπ, n I}


Trigonometric Identity:

A trigonometric equation that holds good for every angle is called a trigonometric identity. Some of the important trigonometric identities are listed below:

  • tan θ = cot θ – 2cot 2θ

  • sin θ sin (60° - θ) sin (60° + θ) = ¼ sin 3θ

  • cos θ cos (60° - θ) cos (60° + θ) = ¼ cos 3θ

  • tan θ tan (60° - θ) tan(60° + θ) = tan 3θ

It is not very easy to solve trigonometric equations every time. At times it becomes very tedious to find the exact value in an equation. Consider tan (x) = 3.2. In such cases inverse trigonometric functions prove useful. Inverse trigonometric functions are the same as the trigonometric functions, except x and y are reversed.

We have also focused on the areas where students generally commit mistakes in the coming sections. Implicit restrictions on the values of various variables are often overlooked even by very vigilant students. Given below is the list of contents covered under this head in the coming sections:

  • Trigonometric Equation and its Solutions

  • Special cases for the Roots

  • System of Trigonometric Equations

  • Important forms of Trigonometric Equations

  • Simultaneous Equations

  • Trigonometric Inequality

  • Inverse circular function

  • Solved Examples

Trigonometry Equation is a vital topic of the IIT JEE Mathematics syllabus. There is a fixed pattern of questions asked form this topic and it is very important to be versant with all the topics as it forms the base for various other topics in various exams. Students are advised to have good understanding of trigonometric functions, trigonometric inequalities and inverse circular functions in order to remain competitive in the JEE.

Example: Solve sin x + √2 = - sin x, x ∈ [0, 2π].

Solution: First, solve for sin x.

The given equation is sin x + √2 = - sin x

This gives 2 sin x = -√2

Hence, sin x = -√2/2

Hence, sin x = -1/√2.

This results in a negative value of sine and sine is negative in Quadrant III and Quadrant IV. 
Moreover, sine attains a value of 1/√2 at an angle of 45°. So we consider the angles of 45° in quadrants 3 and 4.

This gives x = 225° and 315°.

Hence, we get x = 5π/4 or x = 7π/4.  

For more on trigonometric equations, refer the following video

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