A line in the Cartesian plane makes an angle with the positive direction of the -axis

MATHEMATICS · Multi-part · A Level · Mathematics · Coordinate geometry in the (x y) plane · Coordinate geometry in the (x y) plane

Problem

A line in the Cartesian plane makes an angle

\theta

with the positive direction of the

x

-axis, where

0^{\circ}\le \theta<180^{\circ}

. (i) Explain, with reason, why the slope of the line is positive when

0^{\circ}<\theta<90^{\circ}

but negative when

90^{\circ}<\theta<180^{\circ}

. (ii) Without evaluating any trigonometric ratios, state the sign (positive, negative or undefined) of the slope for lines whose inclinations are

47^{\circ},\;95^{\circ},\;173^{\circ}

and

90^{\circ}

. (iii) Give a brief reason why a line whose inclination is

90^{\circ}

does not have a defined slope.