Signs of the trigonometric functions in the third quadrant

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Signs of the trigonometric functions in the third quadrant
Third Quadrant Notation
\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}-




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