Question

If sec Θ =
`(2 √(3) )/(3)` and sin Θ < 0 , then find cot Θ +
`√(2)`

Answer 1

`\sec x=(1)/(\cos x)\\\\\sec\theta=(2\sqrt3)/(3) \ \textgreater \ 0\ and\ \sin\theta \ \textless \ 0\to\theta\ is\ in\ the\ 4-th\ quadrant.\\\theta\in\left((3\pi)/(2);\ 2\pi\right)\\\\\sec\theta=(2\sqrt3)/(3)\\\\(1)/(\cos\theta)=(2\sqrt3)/(3)\Rightarrow\cos\theta=(3)/(2\sqrt3)\cdot(\sqrt3)/(\sqrt3)\\\\\cos\theta=(3\sqrt3)/(2\cdot3)\\\\\cos\theta=(\sqrt3)/(2)\to\theta=(11\pi)/(6)\\\\\cot\theta=\cot(11\pi)/(6)=\cot(5\pi)/(6)=-\sqrt3`


`Answer:\boxed{\sqrt2-\sqrt3}`