First slide

Trigonometric equations

Question

Number of values of $θ$ in $[0, 2 π]$ for which at least two roots of equation
$x^{3} - (1 + \tan θ + \cot θ) x^{2} + (\tan θ \cot θ + \tan θ + \cot θ) x - \tan θ \cot θ = 0$ are equal is

Easy

Solution

Roots are 1, $\tan θ, \cot θ$
$\Rightarrow \tan θ = 1 o r \cot θ = 1 o r \tan θ = \cot θ \Rightarrow \tan θ = \cot θ = 1$
$\Rightarrow θ = \frac{π}{4}, \frac{5 π}{4}$

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