# Fundamental Trigonometric Ratios

1) $Sin\theta = \frac{{perpendicular}}{{hypotenuse}} = \frac{y}{r}$

2) $Cos\theta = \frac{{base}}{{hypotenuse}} = \frac{x}{r}$

3) $Tan\theta = \frac{{perpendicular}}{{base}} = \frac{y}{x} = \frac{{Sin\theta }}{{Cos\theta }}$

4) $Cot\theta = \frac{{base}}{{perpendicular}} = \frac{x}{y} = \frac{{Cos\theta }}{{Sin\theta }}$

5) $Sec\theta = \frac{{hypotenuse}}{{base}} = \frac{r}{x} = \frac{1}{{Cos\theta }}$

6) $Co\sec \theta = \frac{{hypotenuse}}{{perpendicular}} = \frac{r}{y} = \frac{1}{{Sin\theta }}$

7) Signs of trigonometric Ratios
I QUADRANT: $A$, means all trigonometric ratios are positive.
II QUADRANT: $S$, means $Sin$ and $Co\sec$ are positive all others are negative.
III QUADRANT: $T$, means $Tan$ and $Cot$ are positive all others are negative.
IV QUADRANT: $C$, means $Cos$ and $Sec$ are positive all others are negative.

NOTE: (1) Clockwise; we read $ACTS$
(2) Anticlockwise; we read $ASTC$ (All Silver Tea Cups)

 $\theta$ in Quadrant $Sin\theta$ $Cos\theta$ $Tan\theta$ $Cot\theta$ $Sec\theta$ $Co\sec \theta$ $I$ $+$ $+$ $+$ $+$ $+$ $+$ $II$ $+$ $-$ $-$ $-$ $-$ $+$ $III$ $-$ $-$ $+$ $+$ $-$ $-$ $IV$ $-$ $+$ $-$ $-$ $+$ $-$

8) Trigonometric Ratios of Special Angles:

 $\theta$ $Sin\theta$ $Cos\theta$ $Tan\theta$ $Cot\theta$ $Sec\theta$ $Co\sec \theta$ ${0^ \circ }$ $0$ $1$ $0$ $\infty$ $1$ $\infty$ ${30^ \circ }$ $\frac{1}{2}$ $\frac{{\sqrt 3 }}{2}$ $\frac{1}{{\sqrt 3 }}$ $\sqrt 3$ $\frac{2}{{\sqrt 3 }}$ $2$ ${45^ \circ }$ $\frac{1}{{\sqrt 2 }}$ $\frac{1}{{\sqrt 2 }}$ $1$ $1$ $\sqrt 2$ $\sqrt 2$ ${60^ \circ }$ $\frac{{\sqrt 3 }}{2}$ $\frac{1}{2}$ $\sqrt 3$ $\frac{1}{{\sqrt 3 }}$ $2$ $\frac{2}{{\sqrt 3 }}$ ${90^ \circ }$ $1$ $0$ $\infty$ $0$ $\infty$ $1$

RULE: Write $0,1,2,3,4$ divide by $4$ then taking square roots and write the resulting number in column of $Sin\theta$.