Rapsinincke
2022-06-30
Answered

Find the exact value of $\mathrm{cot}({202.5}^{\circ})$

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haingear8v

Answered 2022-07-01
Author has **13** answers

$\mathrm{cot}({202.5}^{0})=\mathrm{cot}({180}^{0}+{22.5}^{0})=\mathrm{cot}({22.5}^{0})$

Now Calculate value of $\mathrm{tan}({22.5}^{0})$

Using

$\mathrm{tan}(2A)=\frac{2\mathrm{tan}A}{1-{\mathrm{tan}}^{2}A}\phantom{\rule{thickmathspace}{0ex}},$

Put $A={22.5}^{0}$

We get

$\mathrm{tan}({45}^{0})=\frac{2\mathrm{tan}({22.5}^{0})}{1-{\mathrm{tan}}^{2}({22.5}^{0})}\Rightarrow 2\mathrm{tan}({22.5}^{0})=1-{\mathrm{tan}}^{2}({22.5}^{0})$

So we get

${\mathrm{tan}}^{2}({22.5}^{0})+2\mathrm{tan}({22.5}^{0})+1=2\Rightarrow {[1+\mathrm{tan}({22.5}^{0})]}^{2}=(\sqrt{2}{)}^{2}$

So we get

$1+\mathrm{tan}({22.5}^{0})=\pm \sqrt{2}$

So we get

$\mathrm{tan}({22.5}^{0})=\sqrt{2}-1$

bcz $\mathrm{tan}({22.5}^{0})>0$

So we get

$\mathrm{cot}({22.5}^{0})=\frac{1}{\mathrm{tan}({22.5}^{0})}=\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}=\sqrt{2}+1$

Now Calculate value of $\mathrm{tan}({22.5}^{0})$

Using

$\mathrm{tan}(2A)=\frac{2\mathrm{tan}A}{1-{\mathrm{tan}}^{2}A}\phantom{\rule{thickmathspace}{0ex}},$

Put $A={22.5}^{0}$

We get

$\mathrm{tan}({45}^{0})=\frac{2\mathrm{tan}({22.5}^{0})}{1-{\mathrm{tan}}^{2}({22.5}^{0})}\Rightarrow 2\mathrm{tan}({22.5}^{0})=1-{\mathrm{tan}}^{2}({22.5}^{0})$

So we get

${\mathrm{tan}}^{2}({22.5}^{0})+2\mathrm{tan}({22.5}^{0})+1=2\Rightarrow {[1+\mathrm{tan}({22.5}^{0})]}^{2}=(\sqrt{2}{)}^{2}$

So we get

$1+\mathrm{tan}({22.5}^{0})=\pm \sqrt{2}$

So we get

$\mathrm{tan}({22.5}^{0})=\sqrt{2}-1$

bcz $\mathrm{tan}({22.5}^{0})>0$

So we get

$\mathrm{cot}({22.5}^{0})=\frac{1}{\mathrm{tan}({22.5}^{0})}=\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}=\sqrt{2}+1$

Cristopher Knox

Answered 2022-07-02
Author has **6** answers

$\mathrm{cot}{202.5}^{0}=\mathrm{cot}22.5$ , in third quadrant. Recognizing half of ${45}^{0}$ you are going to need tan of half angle formula.

Denoting tan full angle by caps, $T={\displaystyle \frac{2t}{1-{t}^{2}}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}t=(\sqrt{1+T}-1),\mathrm{tan}{22.5}^{0}=\sqrt{2}-1$ where we chose positive value from $\pm $ in third quadrant...and the required cot is its reciprocal:

$\mathrm{cot}{22.5}^{0}=\sqrt{2}+1.$

Denoting tan full angle by caps, $T={\displaystyle \frac{2t}{1-{t}^{2}}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}t=(\sqrt{1+T}-1),\mathrm{tan}{22.5}^{0}=\sqrt{2}-1$ where we chose positive value from $\pm $ in third quadrant...and the required cot is its reciprocal:

$\mathrm{cot}{22.5}^{0}=\sqrt{2}+1.$

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