Let and then

Engineering

Mathematics

Inequalities

Introduction to Determinants

Maximum and Minimum Value of Trigonometric Expression

Question

Let $0 < θ < \frac{π}{2}$ and $Δ (x, θ) = |\begin{array}{ccc} x & \tan θ & \cot θ \\ - \tan θ & - x & 1 \\ \cot θ & 1 & x \end{array}|$ then

Δ(0, θ) = 0

$Δ (x, \frac{π}{4}) = x - x^{3}$

Min_0<θ<π/2Δ(1, θ) = 0

Δ(x, θ) is independent of x

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$Δ (0, θ) = ∣ \begin{array}{ccc} 0 & \tan θ & \cot θ \\ - \tan θ & 0 & 1 \\ \cot θ & 1 & 0 \end{array} ∣ = \tan θcot θ - \cot θtan θ = 0$