Question

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

Answer 1

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

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,

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

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬