Consider the following two statements : Statement p : The value of sin 120o can be derived…

Question

Consider the following two statements :

Statement p :

The value of sin 120o can be derived by taking $$\theta  = {240^o}$$  in the equation

2sin$${\theta  \over 2} = \sqrt {1 + \sin \theta }  - \sqrt {1 - \sin \theta } $$

Statement q :

The angles A, B, C and D of any quadrilateral ABCD satisfy the equation

cos$$\left( {{1 \over 2}\left( {A + C} \right)} \right) + \cos \left( {{1 \over 2}\left( {B + D} \right)} \right) = 0$$

Then the truth values of p and q are respectively :

Accepted Answer

Statement p :

sin 120o = cos 30o = $${{\sqrt 3 } \over 2}$$ $$ \Rightarrow $$ 2 sin 120o = $$\sqrt 3 $$

So, $$\sqrt {1 + \sin {{240}^o}}  - \sqrt {1 - \sin {{240}^o}} $$

$$ = \sqrt {{{1 - \sqrt 3 } \over 2}}  - \sqrt {{{1 + \sqrt 3 } \over 2}}  \ne \sqrt 3 $$

Statement q :

So,   A + B + C + D = 2$$\pi $$

$$ \Rightarrow $$ $${{A + C} \over 2} + {{B + D} \over 2} = \pi $$

$$ \Rightarrow $$  cos$$\left( {{{A + C} \over 2}} \right) + \cos \left( {{{B + D} \over 2}} \right)$$

= cos $$\left( {{{A + C} \over 2}} \right)$$ - cos$$\left( {{{A + C} \over 2}} \right) = 0$$

Therefore, statement p is false and statement q is true.

Solutions · 1