Question

If $cosP=\frac{1}{7}$ and $cosQ=\frac{13}{14},$ where P and Q both are acute angles. Then the values of P-Q is

• A. ${30}^{\circ }$
  
• B. ${60}^{\circ }$
  
• C. ${45}^{\circ }$
  
• D. ${75}^{\circ }$

Answer 1

The correct option is B ${60}^{\circ }$


$cosP=\frac{1}{7},cosQ=\frac{13}{14}$
Therefore, $sinP=\sqrt{1-\frac{1}{49}}=\frac{4\sqrt{3}}{7}$
and $Q=\sqrt{1-\frac{169}{196}}=\frac{3\sqrt{3}}{14}$
Hence, $tanP=4\sqrt{3},$ and $tanQ=\frac{3\sqrt{3}}{13}$
cos(P-Q) =cosP cosQ +sinP sinQ
$=\frac{1}{7}\times \frac{13}{14}+\frac{4\sqrt{3}}{7}\times \frac{3\sqrt{3}}{14}$
$=\frac{13+35}{98}$
$=\frac{49}{98}$
$\therefore cos(P-Q)=\frac{1}{2}$
$\Rightarrow P-Q=co{s}^{-1}\frac{1}{2}$
$\Rightarrow P-Q={60}^{\circ }$