Question

evaluate 6cos60°-sin30°+sin²45°+cos²45°​

Answer 1

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

$\mathsf{6\,cos\,60^{\circ}-sin\,30^{\circ}+sin^245^{\circ}+cos^245^{\circ}}$

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

$\textsf{The value of}$

$\mathsf{6\,cos\,60^{\circ}-sin\,30^{\circ}+sin^245^{\circ}+cos^245^{\circ}}$

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

$\mathsf{Consider,}$

$\mathsf{6\,cos\,60^{\circ}-sin\,30^{\circ}+sin^245^{\circ}+cos^245^{\circ}}$

$\textsf{Using values from trigonometry table, we get}$

$\mathsf{=6\left(\dfrac{1}{2}\right)-\dfrac{1}{2}+\left(\dfrac{1}{\sqrt{2}}\right)^2+\left(\dfrac{1}{\sqrt{2}}\right)^2}$

$\mathsf{=3-\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}}$

$\mathsf{=3+\dfrac{1}{2}}$

$\mathsf{=\dfrac{6+1}{2}}$

$\mathsf{=\dfrac{7}{2}}$

$\implies\boxed{\bf\,6\,cos\,60^{\circ}-sin\,30^{\circ}+sin^245^{\circ}+cos^245^{\circ}=\dfrac{7}{2}}$

$\textbf{Standard trigonometric table:}$

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

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