# Determine the exact value of expression. tan(60)times3sin(90)-sin(315)

Question
Trigonometry
Determine the exact value of expression.
$$\displaystyle{\tan{{\left({60}\right)}}}\times{3}{\sin{{\left({90}\right)}}}-{\sin{{\left({315}\right)}}}$$

2020-12-26
Given,
$$\displaystyle{\tan{{\left({60}\right)}}}\times{3}{\sin{{\left({90}\right)}}}-{\sin{{\left({315}\right)}}}$$
Now, $$\displaystyle{\tan{{\left({60}\right)}}}\times{3}{\sin{{\left({90}\right)}}}-{\sin{{\left({315}\right)}}}$$
$$\displaystyle\sqrt{{{3}}}\times{3}{\left({1}\right)}-{\sin{{\left({360}-{45}\right)}}}$$
$$\displaystyle={3}\sqrt{{{3}}}+{\left(-{\sin{{\left({45}\right)}}}\right)}$$
$$\displaystyle={3}\sqrt{{{3}}}+{\frac{{{1}}}{{\sqrt{{{2}}}}}}$$
$$\displaystyle={\frac{{{3}\sqrt{{{3}}}\times\sqrt{{{2}}}+{1}}}{{\sqrt{{{2}}}}}}$$
$$\displaystyle={\frac{{{3}\sqrt{{{6}}}+{1}}}{{\sqrt{{{2}}}}}}$$
$$\displaystyle={\frac{{{3}\sqrt{{{6}}}+{1}}}{{\sqrt{{{2}}}}}}\times{\frac{{\sqrt{{{2}}}}}{{\sqrt{{{2}}}}}}$$
$$\displaystyle={\frac{{{6}\sqrt{{{3}}}+\sqrt{{{2}}}}}{{{2}}}}$$

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