NCERT Class 11-Math՚s: Exemplar Chapter 3 Trigonometric Functions Part 3 (For CBSE, ICSE, IAS, NET, NRA 2023)

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3.1. 9 Trigonometric equations

Equations involving trigonometric functions of a variables are called trigonometric equations. Equations are called identities, if they are satisfied by all values of the unknown angles for which the functions are defined. The solutions of a trigonometric equations for which are called principal solutions. The expression involving integer n which gives all solutions of a trigonometric equation is called the general solution.

General Solution of Trigonometric Equations

(i) If for some angle α, then

, gives general solution of the given equation

(ii) If for some angle α, then

, gives general solution of the given equation

(iii) If hen

, gives general solution for both equations

(iv) The general value of θ satisfying any of the equations and

(v) The general value of θ satisfying equations simultaneously is given by .

(vi) To find the solution of an equation of the form a , we put

Thus, we find

s the solution of the given equation.

Maximum and Minimum values of the expression are and respectively, where A and B are constants.

3.2 Solved Examples

Short Answer Type

Question 1:

A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a hoop whose radius is 48 cm. Find the angle in degrees which is subtended at the centre of hoop.


Given that circular wire is of radius , so when it is cut then its length . Again, it is being placed along a circular hoop of radius . Here, is the length of arc and is the radius of the circle. Therefore, the angle , in radian, subtended by the arc at the centre of the circle is given by

Question 2:

If for all values of θ, then prove that


We have




Question 3:

Find the value of cosec


We have


Question 4:

If lies in the second quadrant, then show that


Given that θ lies in the second quadrant so .

Hence, the required value of the expression is