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IceJOKER [234]
3 years ago
5

URGENT HELP PLEASE!

Mathematics
1 answer:
Vaselesa [24]3 years ago
5 0

Answer:

(a) x=\dfrac{\pi}{8},\dfrac{3\pi}{8},\dfrac{9\pi}{8},\dfrac{11\pi}{8}

(b) x=\dfrac{3\pi}{2},\dfrac{\pi}{6},\dfrac{5\pi}{6}

Step-by-step explanation:

It is given that 0\leq x\leq 2\pi.

(a)

\sqrt{2}\sin 2x=1

\sin 2x=\dfrac{1}{\sqrt{2}}

\sin 2x=\dfrac{\pi}{4}

2x=\dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{9\pi}{4},\dfrac{11\pi}{4}     [\because \sin x=\sin y\Rightarrow x=n\pi+(-1)^ny]

x=\dfrac{\pi}{8},\dfrac{3\pi}{8},\dfrac{9\pi}{8},\dfrac{11\pi}{8}

(b)

\csc^2 x-\csc x-2=0

\csc^2 x-2\csc x+\csc x-2=0

\csc x(\csc x-2)+1(\csc x-2)=0

(\csc x+1)(\csc x-2)=0

\csc x=-1\text{ or }\csc x=2

\sin x=-1\text{ or }\sin x=\dfrac{1}{2}         [\because \sin x=\dfrac{1}{\csc x}]

x=\dfrac{3\pi}{2}\text{ or }x=\dfrac{\pi}{6},\dfrac{5\pi}{6}

Therefore, x=\dfrac{3\pi}{2},\dfrac{\pi}{6},\dfrac{5\pi}{6}.

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