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sin =  \frac{opposite}{hypotenuse}

cos =  \frac{adjacent}{hypotenuse}

tan = \frac{opposite}{adjacent}=\frac{sin}{cos}

Cosecant is the reciprocal of sin.
csc =  \frac{hypotenuse}{opposite}

Secant is the reciprocal of cos.
sec = \frac{hypotenuse}{adjacent}

Cotangent is the reciprocal of tan.
cot = \frac{adjacent}{opposite} = \frac{cos}{sin}

For a circle whose center is the origin and with radius r:

sin =  \frac{y}{r} 

cos =  \frac{x}{r} 

tan =  \frac{y}{x}

csc =  \frac{r}{y}

sec = \frac{r}{x}

cot = \frac{x}{y}