Maximum and Minimum Value:

sin θ = 0 when θ = 0 ˚, 180˚, 360˚ and so on.

Maximum value of sin θ is 1 when θ = 90 ˚

Minimum value of sin θ is –1 when θ = 270 ˚

So, the range of values of sin θ is –1 ≤ sin θ ≤ 1

Maximum value of sinθ = 1

Minimum value of sinθ = – 1 where  0≤ θ ≤ 2π where π = 180º

Maximum value of  asinθ = a

Minimum value of a sinθ = – a where  0≤ θ ≤ 2π where π = 180º, and a constant.

cos θ = 0 when θ = 90 ˚, 270˚ and so on.

Maximum value of cos θ is 1 when θ = 0 ˚, 360˚.

Minimum value of cos θ is –1 when θ = 180 ˚.

So, the range of values of cos θ is – 1 ≤ cos θ ≤ 1.

Maximum value of cosθ = 1

Minimum value of cosθ = – 1 where  0≤ θ ≤ 2π where π = 180º

Maximum value of b cosθ = b

Minimum value of bcosθ = – b where  0≤ θ ≤ 2π where π = 180º and b is constant.

tan θ = 0 when θ = 0 ˚, 180˚, 360˚. tan θ = 1 when θ = 45 ˚ and 225˚.

tan θ = –1 when θ = 135 ˚ and 315˚.

tan θ does not have any maximum or minimum values. The range of values of tan θ is – ∞ < tan θ < ∞

Type I

Maximum value of asinθ + bcosθ = √ (a² + b²)

Minimum value of asinθ + bcosθ = -√ (a² + b²)

Example  – Find the minimum and maximum value of 3 sinθ + 4 cosθ ?

Ans-Minimum value  = – √ (3² + 4²) = -5
Maximum value = √ (3² + 4²) = 5

Type II-

Find the minimum and maximum value of (sinθ.cosθ)ⁿ  ?

Ans- Minimum value = (1/2)ⁿ
The maximum value can go up to infinity.

Example – Find the minimum value of sin⁴ θ cos⁴  θ

Minimum value of sin⁴ θ cos⁴  θ = Minimum value of (sinθ.cosθ)⁴= (1/2)⁴ = 1/16.

Type – III

a sin² θ+ b cos² θ

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 sin² θ+ 5 cos²θ

First check, here a < b
So Maximum value = 5 and Minimum value = 3

Type IV: 

a sin²θ + b cosec²θ,       a cos²θ + b sec²θ,       a tan²θ + b cot²θ
Minimum value = 2√(ab)
The maximum value can go up to infinity.

Example: Find the minimum value of 4 cos²θ + 9 sec²θ
Observe the case, so Minimum value = 2√(4 × 9) = 12

Example:

Minimum value of a sec² θ+ b cosec² θ?
We do not have a formula for this, lets continue
The given function can be written as
a (1 + tan²θ) + b (1 + cot²θ)
= a + b + a tan² θ+ b cot²θ
= (a+b) + (a tan²θ + b cot²θ)
Now observe that we have the formula for finding min value of a tan²θ + b cot²θ
So here minimum value is (a+b) + 2√(ab)