Question

Find the value of sin 30° cos 60° + cos 30° sin 60° + sin 45° cos 45°.​

Answer 1

Answer :

3/2 is the correct answer.

Answer 2

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

Given expression is

$\rm :\longmapsto\:sin30\degree \: cos60\degree + cos30\degree \: sin60\degree + sin45\degree \: cos45\degree$

We know, From Trigonometric ratios of Standard angles, we have

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

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

$\boxed{\tt{ cos30\degree = \frac{ \sqrt{3} }{2} \: }}$

$\boxed{\tt{ cos60\degree = \frac{1}{2} \: }}$

$\boxed{\tt{ cos45\degree = sin45\degree = \frac{1}{ \sqrt{2} }}}$

Now, On substituting the values in given expression, we get

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

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

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

$\rm \: = \: \dfrac{6}{4}$

$\rm \: = \: \dfrac{3}{2}$

Hence,

$\boxed{\tt{ sin30\degree cos60\degree + cos30\degree sin60\degree + sin45\degree cos45\degree =\frac{3}{2}}}$

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