Question

Let A = {$\frac{1967+1686 i\,sin\theta}{7-3i\,cos\theta}:\theta\in R$ }. If A contains exactly one positive integer n, then the value of n is ______

Answer 1

$\frac{281.(7)+218.(6i\,sin\,(x)}{7-3i.(cos\,x)}$

$\Rightarrow 281(\frac{7+6i\,sin\,x}{7-3i\,cos\,x})$

Multiplying both the numerator and denominator by $7+3i\,sin\,x$,

$\Rightarrow 281(\frac{49+21i(cos\,x+2\,sin\,x)+18\,sin\,x\,cos\,x}{49+9cos^2x})$

$\Rightarrow (\frac{49+18\,sin\,x\,cos\,x}{49+9\,cos^2x}+i\frac{21(cos\,x+2\,sin\,x)}{49+9cos^2x})$

Hence we can say that,

$\frac{21\,cos\,x+2\,sin\,x}{49+9\,cos^2\,x}=0$

Hence, the value of n will be 281.