First slide

Extreme values and Periodicity of Trigonometric functions

Question

If $D = |\begin{array}{ccc} 1 & \cos θ & 1 \\ - \sin θ & 1 & - \cos θ \\ - 1 & \sin θ & 1 \end{array}|$ then D lies in the interval

Moderate

A

[0, 4]
B

[2, 4]
C

$[2 - \sqrt{2}, 2 + \sqrt{2}]$
D

[-2, 2]

Solution

Expanding D, we get
$\begin{array}{l} D = 1 + \sin θ \cos θ - \cos θ (- \sin θ - \cos θ) + (- \sin^{2} θ + 1) \\ = 2 + \sin 2 θ + \cos 2 θ = 2 + \sqrt{2} \cos (2 θ - π / 4) \end{array}$
As $- 1 \leq \cos (2 θ - π / 4) \leq 1, 2 - \sqrt{2} \leq D \leq 2 + \sqrt{2}$

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