Question

evaluate;
(sin30 + tan45 - cosec60) / (sec30 + cos60 + cot45) {ans (43 - 24root3) / 11}​

Answer 1

Answer:-

Given that,

$\sf \longmapsto \: \dfrac{sin30 + tan45 - cosec60}{sec30 + cos60 + cot45} \\ \\ \sf \longmapsto \: \dfrac{ \dfrac{1}{2} + 1 - \dfrac{ - 2}{ \sqrt{3} } }{ \dfrac{2}{ \sqrt{3} } + \dfrac{1}{2} + 1} \\ \\ \sf \longmapsto \: \dfrac{ 2 \sqrt{3} - 4 + \sqrt{3} }{4 + 2 \sqrt{3} + \sqrt{3} } \\ \\ \tt \small{\because By \: rationalising \: the \: denomenator,} \\ \sf \longmapsto \: \dfrac{3 \sqrt{3} - 4 }{3 \sqrt{3 + 4} } \times \dfrac{3 \sqrt{3} - 4}{3 \sqrt{3 - 4} } \\ \\ \sf \longmapsto \: \dfrac{27 - 12 \sqrt{3} - 12 \sqrt{3} + 16 }{27 - 16} \\ \\ \sf \longmapsto \: \dfrac{43 - 12 \sqrt{3} - 12 \sqrt{3} }{11} \\ \\ \sf \longmapsto \boxed{\boxed{ \bf \purple{ \dfrac{43 - 24 \sqrt{3} }{11} }}}$

Important formulas:-

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 65^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

Answer 2

Question :-

• Evaluate $\sf{\:\dfrac{sin30° + tan45° - cosec60°}{sec30° + cos60° + cot45°}}$

$\:$

Solution :-

$\sf{⇒\dfrac{sin30° + tan45° - cosec60°}{sec30° + cos60° + cot45°}}$

$\sf{=\dfrac{\dfrac{1}{2} + 1 - \dfrac{2}{\sqrt{3}}}{ \dfrac{2}{\sqrt{3}}+ \dfrac{1}{\sqrt{2}} + 1}}$

$\sf{=\dfrac{\dfrac{\sqrt{3} +2\sqrt{3} -4}{\cancel{2\sqrt{3}}}}{\dfrac{\sqrt{3} +2\sqrt{3} +4}{\cancel{2\sqrt{3}}}}}$

$\sf{=\dfrac{3\sqrt{3} - 4}{3\sqrt{3} + 4}\times \dfrac{3\sqrt{3} - 4}{3\sqrt{3} - 4}}$

$\sf{=\dfrac{(3\sqrt{3})²+ (4)²-2(3\sqrt{3})(4)}{(3\sqrt{3})²-(4)²}}$

$\sf{=\dfrac{27+ 16-24\sqrt{3}}{27-16}}$

$=$ $\bold{\red{\dfrac{43-24\sqrt{3}}{11}}}$

$\:$

Table to remember :-

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}\end{gathered}$