Minimum and Maximum values of Trigonometric Functions

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 a 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

*Note: In these formulae, reciprocal of one another is there.