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General Solution of sin theta = sin alpha

The number of...

Question

The number of solutions of the equation $sin (\frac{π x}{2 \sqrt{3}}) = x^{2} - 2 \sqrt{3} x + 4$

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Solution

$\begin{matrix} sin (\frac{π x}{2 \sqrt{3}}) = x^{2} - 2 \sqrt{3} x + 4 M i n v a l u e o f R H S i s \frac{4 a c - b^{2}}{4 a} = \frac{4 (1) (4) - {(2 \sqrt{3})}^{2}}{4 (1)} = 1 w h i c h o c c u r s a t x = \frac{- b}{2 a} = \frac{2 \sqrt{2}}{2} = \sqrt{3} O n t h e o t h e r h a n d, L . H . S = sin (\frac{π}{2 \sqrt{3}}) h a s max . v a l u e 1 ∴ L . H . S \leq 1 a n d R . H . S \geq 1 \Rightarrow L H S = R H S = 1 w h i c h i s p o s s i b l e o n l y w h e n x = \sqrt{3} ∴ T h e r e i s o n l y o n e r e a l s o l u t i o n \end{matrix}$

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General Solution of sin theta = sin alpha

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