Minimum and Maximum values of Trigonometric Functions

Hello Readers,
Here we are providing you with the tricks to find the minimum and maximum values of Trigonometric Identities which are useful in SSC, Railways and other exams.

Type I: a sinɸ ± b cosɸ,       a sinɸ ± b sinɸ,       a cosɸ ± b cosɸ
Maximum value = √ (a² + b²)
Minimum value  = – √ (a² + b²)

Example: Find the minimum and maximum value of 3 sinɸ + 4 sinɸ
Minimum value  = – √ (3² + 4²) = -5
Maximum value = √ (3² + 4²) = 5

Type 2: (sinɸ cosɸ)ⁿ
Minimum value = (1/2)ⁿ
The maximum value can go up to infinity.

Example: Find the minimum value of sin⁴ɸ cos⁴ɸ
Minimum value = (1/2)⁴ = 1/16

Type 3: a sin²ɸ + b cos²ɸ
If a > b, Maximum value = a and Minimum value = b
If a < b, Maximum value = b and Minimum value = a

Example: Find the minimum and maximum values of 3 sin²ɸ + 5 cos²ɸ
First check, here a < b
So Maximum value = 5 and Minimum value = 3

*Note: You do not have to learn this formula, just observe here that if the equation is of type a sin²ɸ + b cos²ɸ, no matter what, the maximum value is the larger of values (a, b) and minimum value is smaller of values(a, b).

Type 4: a sin²ɸ + b cosec²ɸ,       a cos²ɸ + b sec²ɸ,       a tan²ɸ + b cot²ɸ
Minimum value = 2√(ab)
The maximum value can go up to infinity.

Example: Find the minimum value of 4 cos²ɸ + 9 sec²ɸ
Observe the case, so Minimum value = 2√(4*9) = 12