♦ 1 ≤ sin θ ≤ 1
♦ 1 ≤ cos θ ≤ 1
♦ -∞ < tan θ < ∞
♦ -∞ < cot θ < ∞
♦ sec θ ≥ 1 (when the value of sec θ is positive) or sec θ ≤ -1 (when the value of sec θ is negative)
♦ cosec θ ≥ 1 (when the value of cosec θ is positive) or cosec θ ≤ -1 (when the value of cosec θ is negative)
♦ \(0 ≤ sin^2 θ ≤ 1\)
♦ \(0 ≤ cos^2 θ ≤ 1\)
♦ \(sin^2 θ ≤ |sin θ |≤ 1\)
♦ \( cos^2 θ ≤ |cos θ |≤ 1\)
♦ \(sin^{2a}θ+cos^{2b}θ≤1\) [note: power should be even number]
♦ Maximum value of a sinθ ± b cosθ OR a sinθ ± b sinθ OR a cosθ ± b cosθ is \(\sqrt{a^2+b^2}\)
♦ Minimum value of a sinθ ± b cosθ OR a sinθ ± b sinθ OR a cosθ ± b cosθ is \(-\sqrt{a^2+b^2}\)
♦ Maximum value of \(sin^nθ.cos^nθ \space is \space{1\over 2^n}\)
♦ Minimum value of \(a \space sec^2θ\space +b\space cosec^2θ \space \) OR \({a \over cos^2θ\space} +{b \over sin^2θ}\) is \(({\sqrt{a}}+{\sqrt{b}})^2\) [maximum value depends on θ]
♦ Minimum Value of any equation of type \(ax^n + {b\over x^n } \space is \space 2\sqrt{ab}\) [derived from the concept of Arithmetic mean & Geometric means] [x can be any trigonometric ratio]
Let a ,b are any two real numbers then,
Arithmetic Mean (A.M.) = (a + b) / 2 and
Geometric Mean (G.M.) = √ (a.b)
concept
A.M. ≥ G.M. i.e. (a + b) / 2 ≥ √ (a.b) [you can check for any value of a & b]
a + b ≥ 2√a.b ; In native language we can say value of any equation of form a + b is always greater than or equal to 2√a.b. Hence minimum value of a + b = 2√a.b
Now the question arises how this concept is used in the Trigonometric equation?
► We will try to understand this concept with an example:-
# What is the minimum value of 4sinθ + 9cosecθ ?
Sol:- let us assume a = 4sinθ & b = 9cosecθ
A.M. = (4sinθ + 9cosecθ)/2 & G.M. = √(4sinθ.9cosecθ)
now A.M. ≥ G.M., therefore (4sinθ + 9cosecθ)/2 ≥ √(4sinθ.9cosecθ)since sinθ = 1/cosecθ,
(4sinθ + 9cosecθ) ≥ 2 √(36)
(4sinθ + 9cosecθ) ≥ 12 Hence minimum value of (4sinθ + 9cosecθ) is 12
Imp Point about the concept in relation to trigonometric equations
This concept can only be used to calculate the minimum value of the equation of type \(ax \space + {b\over x}\) where a and b are numerical constant; if we discuss in relation to trigonometric equations, variable x can be any trigonometric ratio and the first trigonometric ratio is opposite of the second or vice-versa like tanθ = 1/cotθ , sinθ = 1/cosecθ , cos2θ = 1/sec2θ.
means equation of type: a.sinθ + b.cosecθ OR a.secθ + b.cosθ OR a.cotθ + b.tanθ & so on. We should note that there should be only an addition (+) sign between them and we can only calculate the minimum value as the maximum value is infinite.
♦ Minimum value of any trigonometric equation as stated above is √ (a.b)