Question

Cos30° + sin60° / 1 + cos60° + sin30° =

Answer 1

Given:

$\rm \to \dfrac{ \cos 30° + \sin 30 °}{1 + \cos 60° + \sin 30°}$

Find:

$\rm Evaluate: \dfrac{ \cos 30° + \sin 30 °}{1 + \cos 60° + \sin 30°}$

Solution:

we, know that

$\rm \to\cos 30° = \sin 60 ° = \frac{ \sqrt{3} }{2}$

$\rm and\cos 60° = \sin 30 ° = \frac{1}{2}$

$\rm Substitute \: these \: values \: in \: \dfrac{ \cos 30° + \sin 30 °}{1 + \cos 60° + \sin 30°}$

So,

$\rm \implies \dfrac{ \cos 30° + \sin 30 °}{1 + \cos 60° + \sin 30°}$

$\rm \implies \dfrac{ \dfrac{ \sqrt{3} }{2} + \dfrac{ \sqrt{3} }{2} }{1 + \dfrac{1}{2} + \dfrac{1}{2} }$

$\rm \implies \dfrac{ \dfrac{ \sqrt{3} + \sqrt{3} }{2} }{\dfrac{2 + 1 + 1}{2} }$

$\rm \implies \dfrac{ \dfrac{ \sqrt{3} + \sqrt{3} }{2} }{\dfrac{4}{2} }$

$\rm \implies \dfrac{2\sqrt{3}}{2} \times \dfrac{2}{4}$

$\rm \implies \sqrt{3} \times \dfrac{1}{2}$

$\rm \implies \dfrac{ \sqrt{3} }{2}$

$\rm Hence, \dfrac{ \cos 30° + \sin 30 °}{1 + \cos 60° + \sin 30°} = \dfrac{ \sqrt{3} }{2}$

Answer 2

GIVEN :-

• (cos30° + sin60°)/(1 + cos60° + sin30°)

TO FIND :-

• Value of (cos30° + sin60°)/(1 + cos60° + sin30°).

SOLUTION :-

$: \implies \displaystyle \sf \dfrac{ (\cos30 ^{ \circ} +\sin60 ^{ \circ}) }{(1 +\cos60 ^{ \circ} +\sin30 ^{ \circ}) }$

$\begin{gathered}\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}\end{gathered}$

$\\ : \implies \displaystyle \sf \frac{ \bigg( \dfrac{ \sqrt{3} }{2} + \dfrac{ \sqrt{3} }{2} \bigg) }{ \bigg(1 + \dfrac{1}{2} + \dfrac{1}{2} \bigg) }$

$: \implies \displaystyle \sf \frac{ \bigg(\dfrac{ \sqrt{3} + \sqrt{3} }{2} \bigg) }{ \bigg(\dfrac{2 + 1 + 1}{2} \bigg)}$

$: \implies \displaystyle \sf \frac{ \bigg(\dfrac{2 \sqrt{3} }{2} \bigg)}{ \bigg( \dfrac{4}{2} \bigg) }$

$: \implies \displaystyle \sf \frac{2 \sqrt{3} }{2} \times \frac{2}{4}$

$: \implies \displaystyle \sf \frac{2 \sqrt{3} }{4}$

$: \implies \underline{ \boxed{ \displaystyle \dfrac{\sqrt{3}}{2} \sf }}$

$\therefore \underline{\displaystyle \sf Value \ of \ \frac{ (\cos30 ^{ \circ} +\sin60 ^{ \circ}) }{(1 +\cos60 ^{ \circ} +\sin30 ^{ \circ}) } = \dfrac{\sqrt{3}}{2} }$