solve ( 4 sin ^ { 2 } ( x ) - 5 sin ( x ) - 6 = 0 ) for all solutions ( 0 leq x < 2 pi )\n\ngive your…

Explanation:

Step1: Let ( t = \sin(x) )

The equation ( 4\sin^{2}(x)-5\sin(x)-6 = 0 ) becomes ( 4t^{2}-5t - 6=0 ).

Step2: Use the quadratic formula ( t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} )

For ( 4t^{2}-5t - 6 = 0 ), where ( a = 4 ), ( b=-5 ), ( c=-6 ). [ \begin{align*} t&=\frac{5\pm\sqrt{(-5)^{2}-4\times4\times(-6)}}{2\times4}\ &=\frac{5\pm\sqrt{25 + 96}}{8}\ &=\frac{5\pm\sqrt{121}}{8}\ &=\frac{5\pm11}{8} \end{align*} ]

Step3: Find the values of ( t )

( t_{1}=\frac{5 + 11}{8}=\frac{16}{8}=2 ), ( t_{2}=\frac{5-11}{8}=\frac{-6}{8}=-\frac{3}{4} )

Step4: Substitute back ( t=\sin(x) )

Since ( - 1\leqslant\sin(x)\leqslant1 ), ( \sin(x)=2 ) has no solution. For ( \sin(x)=-\frac{3}{4} ), ( x=\sin^{-1}\left(-\frac{3}{4}\right)) or ( x=\pi-\sin^{-1}\left(-\frac{3}{4}\right)) ( x = 2\pi-\sin^{-1}\left(\frac{3}{4}\right)\approx2\pi - 0.85 = 5.43) (using ( \sin^{-1}\left(\frac{3}{4}\right)\approx0.85 )) ( x=\pi+\sin^{-1}\left(\frac{3}{4}\right)\approx\pi + 0.85=3.99)

Answer:

(3.99,5.43)