Notes: `15^@` Find the exact values of the sine, cosine, and tangent of the angle.

Wednesday, 13 December 2017

$\displaystyle{{15}}^{\circ}$ Find the exact values of the sine, cosine, and tangent of the angle.

You need to find the values of the sine, cosine and tangent of $\displaystyle{{15}}^{{o}},$ such that:

$\displaystyle{{\sin{{15}}}}^{{o}}={\sin{{\left(\frac{{{{30}}^{{o}}}}{{2}}\right)}}}=\sqrt{{\frac{{{1}-{{\cos{{30}}}}^{{o}}}}{{2}}}}$

$\displaystyle{{\sin{{15}}}}^{{o}}=\sqrt{{\frac{{{2}-\sqrt{{3}}}}{{4}}}}$

$\displaystyle{{\sin{{15}}}}^{{o}}=\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}$

$\displaystyle{{\cos{{15}}}}^{{o}}={\cos{{\left(\frac{{{{30}}^{{o}}}}{{2}}\right)}}}=\sqrt{{\frac{{{1}+{{\cos{{30}}}}^{{o}}}}{{2}}}}$

$\displaystyle{{\cos{{15}}}}^{{o}}=\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{{{\sin{{15}}}}^{{o}}}}{{{{\cos{{15}}}}^{{o}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}}}{{\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}}}$

...

You need to find the values of the sine, cosine and tangent of $\displaystyle{{15}}^{{o}},$ such that:

$\displaystyle{{\sin{{15}}}}^{{o}}={\sin{{\left(\frac{{{{30}}^{{o}}}}{{2}}\right)}}}=\sqrt{{\frac{{{1}-{{\cos{{30}}}}^{{o}}}}{{2}}}}$

$\displaystyle{{\sin{{15}}}}^{{o}}=\sqrt{{\frac{{{2}-\sqrt{{3}}}}{{4}}}}$

$\displaystyle{{\sin{{15}}}}^{{o}}=\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}$

$\displaystyle{{\cos{{15}}}}^{{o}}={\cos{{\left(\frac{{{{30}}^{{o}}}}{{2}}\right)}}}=\sqrt{{\frac{{{1}+{{\cos{{30}}}}^{{o}}}}{{2}}}}$

$\displaystyle{{\cos{{15}}}}^{{o}}=\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{{{\sin{{15}}}}^{{o}}}}{{{{\cos{{15}}}}^{{o}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}}}{{\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{{\left(\sqrt{{{2}-\sqrt{{3}}}}\right)}}}{{{\left(\sqrt{{{2}+\sqrt{{3}}}}\right)}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{{\left(\sqrt{{{4}-{3}}}\right)}}}{{{2}+\sqrt{{3}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{1}}{{{2}+\sqrt{{3}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{1}}{{{2}+\sqrt{{3}}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}=\frac{{{2}-\sqrt{{3}}}}{{{4}-{3}}}$

$\displaystyle{{\tan{{15}}}}^{{o}}={\left({2}-\sqrt{{3}}\right)}$

Hence, evaluating the values of sine, cosine and tangent of $\displaystyle{{15}}^{{o}}$ , yields $\displaystyle{{\sin{{15}}}}^{{o}}=\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}},{{\cos{{15}}}}^{{o}}=\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}},{{\tan{{15}}}}^{{o}}={\left({2}-\sqrt{{3}}\right)}.$

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