Maximum and Minimum Value of Trigonometric Identities

 Maximum and Minimum Value of Trigonometric Identities

Type -1

Example -  (1)Find the maximum and Minimum Value of 3 sin ²x + 4 cos ²x

Sol- Here the 4> 3 so 

•  Maximum Value = 4
• Minimum Value = 3

Example - (2) Find the maximum and Minimum Value of 5 sin ²x + 3 cos ²x

Sol - Here 5>3

• Maximum Value = 5
• Minimum Value = 3

Type -2

Example - (1) Find the Maximum and Minimum Value of 3 sin x + 4 cos y

Sol- If you find the question of this kind, apply the above formulae.

• Maximum Value = √ 9 + 16   = √ 25   = 5
• Minimum Value = - √ 9 + 16   = - √ 25   =  - 5

(2) Find the Maximum and Minimum Value of 3 sin x + 2 cos y

Sol- If you find the question of this kind, apply the above formulae.

• Maximum Value = √ 9 + 4   = √ 13   
• Minimum Value = - √ 9 + 4   = - √ 13   

Type-3

In case of sec²x, cosec²x, cot²x and tan²x, we cannot find the maximum value because they can have infinity as their maximum value. So in question containing these trigonometric identities, you will be asked to find the minimum values only. The typical question forms are listed below:

Example- (1) Find the Minimum value of 3 sin ²x + 4 cosec ²x
sol - this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

• Minimum Value = 2√ 3 x  4= 2√ 12 

Example (2) Find the Minimum value of 9 cos ²x + 2 sec ²x
sol - this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

• Minimum Value = 2√ 9 x  2= 2√ 18 

Example (3) -Find the Minimum value of 8 tan ²x + 7 cot ²x
sol - this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

• Minimum Value = 2√ 8 x  7= 2√ 56

Type-4

In this type, we will give you the explanation of question which are different from type-1, type-2 and type-3. If you find this kind of questions, you will have to convert these questions into type-1,2 or 3 by  using trigonometric formulas 

Example - Find the Minimum Value of  Sec ²x +  cosec ²x

Sol - 1 + tan ²x +  cosec ²x    --------------------------------------------(Sec ²x = 1 + tan ²x)

= 1+ tan ²x + 1 +  cot ²x ------------------------------------------------(cosec ²x = 1 + cot ²x )

=2 + tan ²x +  cot ²x---------------------------------------------------apply type-3 formula

=2 + 2 √ 1 x  1

= 2 + 2
=4