Question

evaluate : sin60° + cos30° + sin30° cos60° ​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{sin\,60^\circ+cos\,30^\circ+sin\,30^\circ\,cos60^\circ}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{sin\,60^\circ+cos\,30^\circ+sin\,30^\circ\,cos60^\circ}$

$\underline{\textbf{Solution:}}$

$\underline{\textsf{Concept used:}}$

$\textbf{Standard trigonometric table:}$  

$\left\begin{array}{|c|c|c|c|c|c|}\cline{1-6}&0^{\circ}&30^{\circ}&45^{\circ}&60^{\circ}&90^{\circ}\\\cline{1-6}\bf\,sin\theta&0&\frac{1}{2}&\frac{1}{\sqrt2}&\frac{\sqrt3}{2}&1\\\cline{1-6}\bf\,cos\theta&1&\frac{\sqrt3}{2}&\frac{1}{\sqrt2}&\frac{1}{2}&0\\\cline{1-6}\bf\,tan\theta&0&\frac{1}{\sqrt3}&1&\sqrt3&\infty\\\cline{1-6}\end{array}\right$  

$\mathsf{Consider,}$

$\mathsf{sin\,60^\circ+cos\,30^\circ+sin\,30^\circ\,cos60^\circ}$

$\mathsf{=\dfrac{\sqrt3}{2}+\dfrac{\sqrt3}{2}+\dfrac{1}{2}{\times}\dfrac{1}{2}}$

$\mathsf{=\dfrac{\sqrt3+\sqrt3}{2}+\dfrac{1}{4}}$

$\mathsf{=\dfrac{2\sqrt3}{2}+\dfrac{1}{4}}$

$\mathsf{=\sqrt3+\dfrac{1}{4}}$