Question

# The equation $$\displaystyle x^{3} - \frac{3}{4} x = - \frac{\sqrt{3}}{8}$$ is satisfied by

A
x=cos(5π18)
B
x=cos(7π18)
C
x=cos(23π18)
D
x=cos(17π18)

Solution

## The correct options are A $$\displaystyle x = \cos \left ( \frac{5 \pi}{18} \right )$$ C $$\displaystyle x = \cos \left ( \frac{7 \pi}{18} \right )$$ D $$\displaystyle x = \cos \left ( \frac{17 \pi}{18} \right )$$$$\displaystyle { x }^{ 3 }-\frac { 3 }{ 4 } x=-\frac { \sqrt { 3 } }{ 8 }$$Let $$x\Rightarrow \cos { \theta }$$$$\displaystyle \cos ^{ 3 }{ \theta } -\frac { 3 }{ 4 } \cos { \theta } =\frac { -\sqrt { 3 } }{ 8 }$$$$\displaystyle \Rightarrow 4\cos ^{ 3 }{ \theta } -3\cos { \theta } =\frac { -\sqrt { 3 } }{ 2 }$$$$\displaystyle \cos { 3\theta } =\frac { -\sqrt { 3 } }{ 2 } \Rightarrow 3\theta =2n\pi +\frac { 5\pi }{ 6 }$$ or $$\displaystyle 2n\pi +\frac { 7\pi }{ 6 }$$$$\displaystyle \Rightarrow \theta =\frac { 1 }{ 18 } \left( 12\pi n\pm 5\pi \right)$$$$\displaystyle \therefore \theta =\frac { 5\pi }{ 18 } ,\frac { 7\pi }{ 18 } ,\frac { 17\pi }{ 18 }$$Maths

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