Signs of the trigonometric functions in the third quadrant

Name

Signs of the trigonometric functions in the third quadrant
$\text{P is any point in the third quadrant}\,$

$\overline {ON}\text{ x-coordinate or abscissa of point P, negative}\,$

$\overline {OM}\text{ y-coordinate or ordinate of point P, negative}\,$

$\overline {OP}\text{ distance from the origin, always}\,$

$\text{positive because it is a length}\,$

$\overline {OM}=\overline{NP}\,$

$\sin \alpha=\frac{\overline{NP}}{\overline {OP}}=\frac{\text{y-coordinate}}{\text{distance from the origin}}=\frac{-a}{b}\qquad \color{Red}-$
$\cos \alpha=\frac{\overline{ON}}{\overline {OP}}=\frac{\text{x-coordinate}}{\text{distance from the origin}}=\frac{-c}{b}\qquad \color{Red}-$
$\tan \alpha=\frac{\overline{NP}}{\overline {ON}}=\frac{\text{y-coordinate}}{\text{x-coordinate}}=\frac{-a}{-c}\qquad \color{Red}+$
$\cot \alpha=\frac{\overline{ON}}{\overline {NP}}=\frac{\text{x-coordinate}}{\text{y-coordinate}}=\frac{-c}{-a}\qquad \color{Red}+$
$\sec \alpha=\frac{\overline{OP}}{\overline {ON}}=\frac{\text{distance from the origin}}{\text{x-coordinate}}=\frac{b}{-c}\qquad \color{Red}-$
$\csc \alpha=\frac{\overline{OP}}{\overline {NP}}=\frac{\text{distance from the origin}}{\text{y-coordinate}}=\frac{b}{-a}\qquad \color{Red}-$

Usage

• Write here usage for the table