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Trigonometrical Equations
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The solution set of the system of equation
$x\,\, + \,\,y\,\, = \,\,\frac{{2\pi }}{3},\,{\rm{cos}}\,{\rm{x + }}\,{\rm{ cos}}\,{\rm{y}}\,{\rm{ = }}\,\frac{3}{2},$ where $x$ and $y$ are real in
A
a finite non-empty set
B
null set
C
$\infty $
D
none of these
Solution
$\cos x+\cos y=\frac{3}{2}$
$\Rightarrow \quad 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}=\frac{3}{2}$
$\Rightarrow \quad 2 \cos \frac{\pi}{3} \cos \frac{x-y}{2}=\frac{3}{2} \quad\left(\because x+y=\frac{2 \pi}{3}\right)$
$\Rightarrow \quad \cos \frac{x-y}{2}=\frac{3}{2}$
$\text { which is not possible } \quad\left(\because \frac{3}{2}>1\right)$
Hence the system of equations has no solution.
Standard 11
Mathematics
