Question

Find the values of:
(1) sina 60° - cos2 45° + 3 tan2 30°​​

Answer 1

Answer:

Given :-

• Find the value of:
• $sin \: {60}^{0} - {cos}^{2} {45}^{0} + 3 {tan}^{2} {30}^{0}$

To prove :-

• We should find trigonometric equation values.

Explanation :-

• We know that,

Trigonometric identities: -

• $sin {60}^{0} = \frac{ \sqrt{3} }{ \: \: \: 2}$

• $cos {45}^{0} = \frac{1}{ \sqrt{2} }$

• $tan {30}^{0 } = \frac{1}{ \sqrt{3} }$

Now,

• Applying the given values we get that,

• $\frac{ \sqrt{ {(3)}^{2} } }{ {2}^{2} } - \frac{1}{ \sqrt{ {2}^{2} } } + 3 \times \frac{1}{ \sqrt{ {3}^{2} } }$

• $= \frac{3}{4} + \frac{1}{2}$

• $= \frac{5}{4}$

Therefore,

• The value of trigonometric equation we is 5/4.

Hope it helps you.

Thank you ..

Answer 2

Appropriate Question

Find the value of

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

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

Consider,

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

We know that,

$\boxed{ \bf{ \:sin60\degree = \frac{ \sqrt{3} }{2}}}$

$\boxed{ \bf{ \:cos45\degree = \frac{1}{ \sqrt{2} }}}$

and

$\boxed{ \bf{ \:tan30\degree = \frac{1}{ \sqrt{3} }}}$

So, on substituting the values, we get

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

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

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

$\rm \:  =  \: \dfrac{3 - 2 + 4}{4}$

$\rm \:  =  \: \dfrac{1 + 4}{4}$

$\rm \:  =  \: \dfrac{5}{4}$

Hence,

$\rm :\longmapsto\:\boxed{ \bf{ \: {sin}^{2}60 \degree \: - \: {cos}^{2}45\degree \: + \: 3 {tan}^{2}60\degree = \frac{5}{4} }}$

Additional Information :-

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