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[Solved] Trigonometry proof equations

 
Trigonometry 1 & 2
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June 15, 2021 2:32 pm
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(@armor)
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Solve the equation:   sin x + cos x + sin x cos x=1

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June 15, 2021 2:32 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

Multiply both sides of the equation by 2 and adding 1, we obtain \small 2(sinx+cosx)+2sinxcosx+1=3\Rightarrow (sinx+cosx)^2+2(sinx+cosx)-3=0

\small \Rightarrow [(sinx+cosx)-1][(sinx+cosx)+3]=0

\small \Rightarrow sinx+cosx=1   or  \small sinx+cosx=-3 .

Since  \small -1\leqslant sinx\leqslant 1, -1\leqslant cosx\leqslant 1,  we have    \small sinx+cosx\neq -3

\small \Rightarrow    \small sinx+cosx=1\Rightarrow \sqrt{2}sin(x+\frac{\Pi }{4})=1\Rightarrow sin(x+\frac{\Pi }{4})=\frac{\sqrt{2}}{2}

\small \Rightarrow x+\frac{\Pi }{4}=2k\Pi +\frac{\Pi }{4},    or   \small x+\frac{\Pi }{4}=(2k+1)\Pi -\frac{\Pi }{4}\Rightarrow x=2kx,    or  \small x=2kx+\frac{\Pi }{2}, (k\epsilon \mathbb{Z}).

Therefore, the solution of the equation is   \small (x|x=2k\Pi ,k\epsilon \mathbb{Z})\bigcup (x|x=2k\Pi +\frac{\Pi }{2}, k\epsilon \mathbb{Z}) .

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