What is t he exact value of csc(arctan-1/3)) ?

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What is t he exact value of csc(arctan-1/3)) ?

what is t he exact value of csc(arctan-1/3)) ?

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Answer 1 · 2018-05-05T05:39:42

$csc (arctan (- \frac{1}{3})) = - \sqrt{10}$ $csc(arctan(-1/3))=-sqrt10$

Explanation:

Let $arctan (- \frac{1}{3}) = A$ $arctan(-1/3)=A$ , then $tan A = - \frac{1}{3}$ $tanA=-1/3$ and hence $A$ $A$ is in interval $- \frac{π}{2} < A < 0$ $-pi/2< A < 0$ i.e. $A$ $A$ is in $Q 4$ $Q4$

and as we are seeking $csc A$ $cscA$ and it is negative in $Q 4$ $Q4$

As $tan A = - \frac{1}{3}$ $tanA=-1/3$ , $cot A = \frac{1}{- \frac{1}{3}} = - 3$ $cotA=1/(-1/3)=-3$

and $csc A = - \sqrt{1 + {cot}^{2} A} = - \sqrt{1 + {(- 3)}^{2}} = - \sqrt{10}$ $cscA=-sqrt(1+cot^2A)=-sqrt(1+(-3)^2)=-sqrt10$

hence $csc (arctan (- \frac{1}{3}) = - \sqrt{10}$ $csc(arctan(-1/3)=-sqrt10$

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What is t he exact value of csc(arctan-1/3)) ?