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Trigonometric equations

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Question

Number of values of the parameter $α \in [0, 2 π]$ for which the quadratic function, $(\sin α) x^{2} + 2 \cos αx + \frac{1}{2} (\cos α + \sin α)$ is the square of a linear function is

Moderate

Solution

Given expression is a perfect square if Disc = 0
$\begin{array}{l} \Rightarrow 4 \cos^{2} α - 2 (\cos α + \sin α) \sin α = 0 \\ \Rightarrow 4 \cos^{2} α - 2 \sin^{2} α - 2 \sin αcos α = 0 \\ \Rightarrow \tan^{2} α + \tan α - 2 = 0 \\ \Rightarrow (\tan α - 1) (\tan α + 2) = 0 \Rightarrow α = \frac{π}{4} or π - \tan^{- 1} (2) \end{array}$

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