Question

find the value of 3 sin²30+2tan²60-5tan²45.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {3sin}^{2}30 \degree \: + {2tan}^{2}60\degree - 5 {tan}^{2}45\degree$

We know,

$\red{ \boxed{ \sf{ \:sin30\degree = \frac{1}{2} \: }}}$

$\red{ \boxed{ \sf{ \:tan60\degree = \sqrt{3} \: }}}$

and

$\red{ \boxed{ \sf{ \:tan45\degree = 1 \: }}}$

So, on substituting all these values, we get

$\rm \:  =  \: 3 {\bigg[\dfrac{1}{2} \bigg]}^{2} + 2 {\bigg[ \sqrt{3}\bigg]}^{2} - 5 {(1)}^{2}$

$\rm \:  =  \: 3 \times \dfrac{1}{4} + 2 \times 3 - 5 \times 1$

$\rm \:  =  \: \dfrac{3}{4} + 6 - 1$

$\rm \:  =  \: \dfrac{3}{4} + 5$

$\rm \:  =  \: \dfrac{3 + 20}{4}$

$\rm \:  =  \: \dfrac{23}{4}$

Hence,

$\red{\bf\implies \: \boxed{ \sf{ \: {3sin}^{2}30 \degree \: + {2tan}^{2}60\degree - 5 {tan}^{2}45\degree \: }}}$

Additional Information :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$