Business Mathematics and Statistics · Ch 4 — Trigonometry

Trigonometric Ratios and Signs in the Four Quadrants

Place an angle $\theta$ in standard position and let $P(x, y)$ be any point (other than the origin) on its terminal side, at distance $r = \sqrt{x^2+y^2}$ from the origin. The six trigonometric ratios of $\theta$ are then defined for ANY angle, not just angles inside a right triangle:

\sin\theta = \frac{y}{r}, \quad \cos\theta = \frac{x}{r}, \quad \tan\theta = \frac{y}{x}, \quad \operatorname{cosec}\theta = \frac{r}{y}, \quad \sec\theta = \frac{r}{x}, \quad \cot\theta = \frac{x}{y}

( $\tan\theta$ and $\sec\theta$ are undefined when $x = 0$ ; $\cot\theta$ and $\operatorname{cosec}\theta$ are undefined when $y = 0$ .) Since $r$ is always taken positive, the sign of each ratio depends only on the signs of $x$ and $y$ — that is, on which quadrant the terminal side falls in.

Quadrant	$x$	$y$	Ratios positive
I ( $0^\circ$ to $90^\circ$ )	$+$	$+$	all six
II ( $90^\circ$ to $180^\circ$ )	$-$	$+$	$\sin\theta, \operatorname{cosec}\theta$
III ( $180^\circ$ to $270^\circ$ )	$-$	$-$	$\tan\theta, \cot\theta$
IV ( $270^\circ$ to $360^\circ$ )	$+$	$-$	$\cos\theta, \sec\theta$

This is often remembered by the mnemonic All – Sin – Tan – Cos (quadrants I, II, III, IV), telling you which ratios are positive in each quadrant; every other ratio in that quadrant is negative. …

Definition 1The Six Trigonometric Ratios

For a point $P(x,y)$ on the terminal side of $\theta$ at distance $r$ from the origin: $\sin\theta=y/r$ , $\cos\theta=x/r$ , $\tan\theta=y/x$ , $\operatorname{cosec}\theta=r/y$ , $\s …

Definition 2Sign Convention (All–Sin–Tan–Cos Rule)

Quadrant I: all ratios positive. Quadrant II: only sin, cosec positive. Quadrant III: only tan, cot positive. Quadrant IV: o …

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