# NCERT Class 11-Math's: Exemplar Chapter 3 Trigonometric Functions Part 3

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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 whichare 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 areand 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.

**Answer:**

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

**Answer:**

We have

Therefore,

Also,

Hence,

**Question 3:**

Find the value of cosec

**Answer:**

We have

cosec

**Question 4:**

If lies in the second quadrant, then show that

**Answer:**

Given that θ lies in the second quadrant so .

Hence, the required value of the expression is