Basics of Trigonometry

$$ 1. \ \sin \theta = \frac {\text{Opposite Side}}{\text{Hypotenuse}} = \frac {AB}{OB} $$

$$ 2. \ \cos \theta = \frac {\text{Adjacent Side}}{\text{Hypotenuse}} = \frac {OA}{OB} $$

$$ 3. \ \tan \theta = \frac {\text{Opposite Side}}{\text{Adjacent Side}} = \frac {AB}{OA} $$

$$ 4. \ \sec \theta = \frac {\text{Hypotenuse}}{\text{Adjacent Side}} = \frac {OB}{AB} $$

$$ 5. \ \text{cosec} \theta = \frac {\text{Hypotenuse}}{\text{Opposite Side}} = \frac {OB}{OA} $$

$$ 6. \ \cot \theta = \frac {\text{Adjacent Side}}{\text{Opposite Side}} = \frac {OA}{AB} $$

Example:

Determine the six trigonometric ratios for angle D in the right triangle below.

$$ \sin \theta = \frac {\text{Opposite Side}}{\text{Hypotenuse}} = \frac 45 $$

$$ \cos \theta = \frac {\text{Adjacent Side}}{\text{Hypotenuse}} = \frac 35 $$

$$ \tan \theta = \frac {\text{Opposite Side}}{\text{Adjacent Side}} = \frac 43 $$

$$ \sec \theta = \frac {\text{Hypotenuse}}{\text{Adjacent Side}} = \frac 54 $$

$$ \text{cosec} \theta = \frac {\text{Hypotenuse}}{\text{Opposite Side}} = \frac 53 $$

$$ \cot \theta = \frac {\text{Adjacent Side}}{\text{Opposite Side}} = \frac 34 $$