Question

What is the value of the following expression:sqrt3cot15^circ

Non-right triangles and trigonometry
ANSWERED
asked 2021-09-13
What is the value of the following expression:
\(\displaystyle\sqrt{{3}}{{\cot{{15}}}^{\circ}}\)

Expert Answers (1)

2021-09-14
First, find \(\displaystyle{{\tan{{15}}}^{\circ}}\)
\(\displaystyle{\tan{{\left({A}-{B}\right)}}}=\frac{{{\tan{{A}}}-{\tan{{B}}}}}{{{1}+{\tan{{A}}}{\tan{{B}}}}}\)
\(\displaystyle{{\tan{{15}}}^{\circ}=}{\tan{{\left({60}^{\circ}-{45}^{\circ}\right)}}}\)
\(\displaystyle{{\tan{{15}}}^{\circ}=}\frac{{{{\tan{{60}}}^{\circ}-}{\tan{{45}}}^{\circ}}}{{{1}+{{\tan{{60}}}^{\circ}{\tan{{45}}}^{\circ}}}}\)
\(\displaystyle{{\tan{{15}}}^{\circ}=}\frac{{\sqrt{{3}}-{1}}}{{{1}+{\left(\sqrt{{3}}\right)}{\left({1}\right)}}}\)
\(\displaystyle=\frac{{\sqrt{{3}}-{1}}}{{\sqrt{{3}}+{1}}}\)
Then,
\(\displaystyle{{\cot{{15}}}^{\circ}=}\frac{{\sqrt{{3}}+{1}}}{{\sqrt{{3}}-{1}}}\)
\(\displaystyle{{\cot{{15}}}^{\circ}=}\frac{{\sqrt{{3}}+{1}}}{{\sqrt{{3}}-{1}}}\times\frac{{\sqrt{{3}}+{1}}}{{\sqrt{{3}}+{1}}}\)
\(\displaystyle{{\cot{{15}}}^{\circ}=}\frac{{{3}+{2}\sqrt{{3}}+{1}}}{{{3}-{1}}}\)
\(\displaystyle{{\cot{{15}}}^{\circ}=}\frac{{{2}\sqrt{{3}}+{4}}}{{2}}\)
\(\displaystyle{c}{\quad\text{or}\quad}{15}^{\circ}=\sqrt{{3}}+{2}\)
Substitute to given expression:
\(\displaystyle\sqrt{{3}}{{\cot{{15}}}^{\circ}=}\sqrt{{3}}{\left(\sqrt{{3}}+{2}\right)}\)
\(\displaystyle\sqrt{{3}}{{\cot{{15}}}^{\circ}=}{3}+{2}\sqrt{{2}}\)
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