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22 votes
22 votes
4)
Solve the following equation for exact solution
sin 2x + sin x + 2 cos x + 1 = 0

User Vaugham
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1 Answer

12 votes
12 votes


\displaystyle\\\\Answer:\ \\x=(3)/(2) \pi +2\pi \mathbb N \\\\x=(2)/(3)\pi +2\pi \mathbb N\\\\x=(4)/(3) \pi +2\pi \mathbb N

Explanation:


\displaystyle\\sin(2x)+sin(x)+2cos(x)+1=0\\\\2sin(x)cos(x)+sin(x)+2cos(x)+1=0\\\\2sin(x)cos(x)+2cos(x)+sin(x)+1=0\\\\2cos(x)(sin(x)+1)+(sin(x)+1)=0\\\\(sin(x)+1)(2cos(x)+1)=0\\\\a)\ sin(x)+1=0\\\\sin(x)=-1\\\\x=(3)/(2) \pi +2\pi \mathbb N\\\\b)\ 2cos(x)+1=0\\\\2cos(x)=-1

Divide both parts of the equation by 2:


\displaystyle\\cos(x)=-(1)/(2) \\\\x=(2)/(3)\pi +2\pi \mathbb N\\\\x=(4)/(3) \pi +2\pi \mathbb N

User Tymel
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