The roots of $5 x^{2} - 7 x + k = 0$ are Sin A and Cos A. Find the value of k.

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Solution

$5 x^{2} - 7 x + k = 0 \begin{matrix} 5 s i n^{2} A - 7 s i n A + k = 0 & k = 7 s i n A - 5 s i n^{2} A \to (1) 5 c o s^{2} A - 7 c o s A + k = 0 & k = 7 c o s A - 5 c o s^{2} A \to (2) \end{matrix}$
(1) =(2)
$7 s i n A - 5 s i n^{2} A = 7 c o s A - 5 c o s^{2} A \Rightarrow s i n A = c o s A \Rightarrow A = 45^{\circ} k = 7 s i n 45^{\circ} - 5 s i n^{2} 45^{\circ} = \frac{7}{\sqrt{2}} - 5 {(\frac{1}{\sqrt{2}})}^{2} k = \frac{7}{\sqrt{2}} - \frac{5}{2} = \frac{14 - 5 \sqrt{2}}{2 \sqrt{2}}$

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