If y= arctan x and x<0 then y lies in what quadrant ?

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- Feb 22nd 2011, 02:02 AMhoanghai549Trigonometry
If y= arctan x and x<0 then y lies in what quadrant ?

- Feb 22nd 2011, 02:14 AMsa-ri-ga-ma
In the co-ordinate axis, if the angle subtended at the origin is θ, then tanθ = y/x. tanθ will be negative when x will be negative of y will be negative. Now decide in which quadrant the angle will be?

- Feb 22nd 2011, 02:31 AMemakarovQuote:

tanθ will be negative when x will be negative of y will be negative.

*and*y > 0 or y < 0*and*x > 0. Also, one must take into account that arctan returns angles between $\displaystyle -\pi/2$ and $\displaystyle \pi/2$. - Feb 22nd 2011, 05:14 AMSoroban
Hello, hoanghai549!

Quote:

$\displaystyle \text{If }y\,=\, \arctan x\,\text{ and }\,x<0,\,\text{ then }y\text{ lies in what quadrant?}$

We have: .$\displaystyle y\,=\,\arctan x \quad\Rightarrow\quad \tan y \,=\,x\;\;\text{ and }x\text{ is negative.}$

The tangent function is negative in Quadrants 2 and 4.

- Feb 22nd 2011, 06:49 AMhoanghai549