Question

The point $A(\sin \theta ,\cos \theta )$ is a unit away from the point B $B(2\cos {75}^{\circ },2\sin {75}^{\circ })if{0}^{\circ }⩽\theta <{360}^{\circ }\theta =$

• A. ${195}^{\circ }$
• B. ${105}^{\circ }$
• C. ${285}^{\circ }$
• D. ${270}^{\circ }$

Answer 1

The correct option is A ${195}^{\circ }$

Solution :

$A(sin\theta ,cos\theta )B(2cos{75}^{\circ },2sin{75}^{\circ })$

Given $AB=1unit.$

$AB=\sqrt{(sin\theta -2cos75{)}^2+(cos\theta -2sin75{)}^2}=1$

$(sin\theta -2cos75{)}^2+(cos\theta -2sin75{)}^2=1$

$si{n}^2\theta +4co{s}^2{75}^{\circ }-4sin\theta cos75+co{s}^2\theta +4si{n}^2{75}^{\circ }-4cos\theta sin75=1$

$1+4-4[sin\theta cos75+cos\theta sin75]=1$

$\Rightarrow sin(\theta +75)=1$

$\Rightarrow \theta +75=\frac{n\pi }{2}$

$\Rightarrow \theta =\frac{n\pi }{2}-75$

$\because 0<\theta <{360}^{\circ }$ $\therefore \theta =15$ or $195$