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 and all others are negative.
III QUADRANT: $$T$$ means $$Tan$$ and $$Cot$$ are positive and all others are negative.
IV QUADRANT: $$C$$ means $$Cos$$ and $$Sec$$ are positive and all others are negative.
I QUADRANT: $$A$$ means all trigonometric ratios are positive.
II QUADRANT: $$S$$ means $$Sin$$ and $$Co\sec $$ are positive and all others are negative.
III QUADRANT: $$T$$ means $$Tan$$ and $$Cot$$ are positive and all others are negative.
IV QUADRANT: $$C$$ means $$Cos$$ and $$Sec$$ are positive and all others are negative.
NOTE: (1) Clockwise we read $$ACTS$$
(2) Counterclockwise we read $$ASTC$$ (All Silver Tea Cups)
(2) Counterclockwise we read $$ASTC$$ (All Silver Tea Cups)
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$$\theta $$ in Quadrant
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$$Sin\theta $$
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$$Cos\theta $$
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$$Tan\theta $$
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$$Cot\theta $$
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$$Sec\theta $$
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$$Co\sec \theta $$
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$$I$$
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$$ + $$
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$$ + $$
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$$ + $$
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$$ + $$
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$$ + $$
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$$ + $$
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$$II$$
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$$ + $$
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$$ – $$
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$$ – $$
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$$ – $$
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$$ – $$
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$$ + $$
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$$III$$
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$$ – $$
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$$ – $$
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$$ + $$
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$$ + $$
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$$ – $$
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$$ – $$
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$$IV$$
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$$ – $$
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$$ + $$
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$$ – $$
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$$ – $$
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$$ + $$
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$$ – $$
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8) Trigonometric Ratios of Special Angles:
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$$\theta $$
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$$Sin\theta $$
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$$Cos\theta $$
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$$Tan\theta $$
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$$Cot\theta $$
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$$Sec\theta $$
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$$Co\sec \theta $$
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$${0^ \circ }$$
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$$0$$
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$$1$$
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$$0$$
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$$\infty $$
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$$1$$
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$$\infty $$
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$${30^ \circ }$$
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$$\frac{1}{2}$$
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$$\frac{{\sqrt 3 }}{2}$$
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$$\frac{1}{{\sqrt 3 }}$$
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$$\sqrt 3 $$
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$$\frac{2}{{\sqrt 3 }}$$
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$$2$$
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$${45^ \circ }$$
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$$\frac{1}{{\sqrt 2 }}$$
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$$\frac{1}{{\sqrt 2 }}$$
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$$1$$
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$$1$$
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$$\sqrt 2 $$
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$$\sqrt 2 $$
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$${60^ \circ }$$
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$$\frac{{\sqrt 3 }}{2}$$
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$$\frac{1}{2}$$
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$$\sqrt 3 $$
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$$\frac{1}{{\sqrt 3 }}$$
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$$2$$
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$$\frac{2}{{\sqrt 3 }}$$
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$${90^ \circ }$$
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$$1$$
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$$0$$
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$$\infty $$
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$$0$$
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$$\infty $$
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$$1$$
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RULE: Write $$0,1,2,3,4$$, divide by $$4$$ and then take square roots and write the resulting number in the column of $$Sin\theta $$.