Difference between revisions of "Relationship between arctan and arccot"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\mathrm{arctan}(z) = \mathrm{arccot}\left( \dfrac{1}{z} \right),$$ | $$\mathrm{arctan}(z) = \mathrm{arccot}\left( \dfrac{1}{z} \right),$$ | ||
where $\mathrm{arctan}$ denotes the [[arctan|inverse tangent]] and $\mathrm{arccot}$ denotes the [[arccot|inverse cotangent]]. | where $\mathrm{arctan}$ denotes the [[arctan|inverse tangent]] and $\mathrm{arccot}$ denotes the [[arccot|inverse cotangent]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | Let $y = \arctan \left( \dfrac{1}{z} \right)$. Then since arctan is the [[inverse function]] of [[tangent]], | |
| − | + | $$\tan(y)=\dfrac{1}{z}.$$ | |
| + | By the definition of cotangent, we get | ||
| + | $$\cot(y)=z.$$ | ||
| + | Since $\mathrm{arccot}$ is the inverse function of $\cot$, take the $\mathrm{arccot}$ of each side to get | ||
| + | $$y = \mathrm{arccot}(z).$$ | ||
| + | Therefore we have shown | ||
| + | $$\arctan \left( \dfrac{1}{z} \right) = \mathrm{arccot}(z),$$ | ||
| + | as was to be shown. | ||
| + | |||
| + | ==References== | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 14:45, 25 September 2016
Theorem
The following formula holds: $$\mathrm{arctan}(z) = \mathrm{arccot}\left( \dfrac{1}{z} \right),$$ where $\mathrm{arctan}$ denotes the inverse tangent and $\mathrm{arccot}$ denotes the inverse cotangent.
Proof
Let $y = \arctan \left( \dfrac{1}{z} \right)$. Then since arctan is the inverse function of tangent, $$\tan(y)=\dfrac{1}{z}.$$ By the definition of cotangent, we get $$\cot(y)=z.$$ Since $\mathrm{arccot}$ is the inverse function of $\cot$, take the $\mathrm{arccot}$ of each side to get $$y = \mathrm{arccot}(z).$$ Therefore we have shown $$\arctan \left( \dfrac{1}{z} \right) = \mathrm{arccot}(z),$$ as was to be shown.
