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NCERT Class 11- Math՚s: Exemplar Chapter 3 Trigonometric Functions Part 3
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.
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