2.1 The Six Trigonometric Ratios

Notes

In a right triangle, label the sides relative to an acute angle $\theta$: opposite, adjacent, and hypotenuse (across from the right angle).

• $\displaystyle \sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\displaystyle \cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\displaystyle \tan\theta = \frac{\text{opp}}{\text{adj}}$ (memory aid: SOH-CAH-TOA).
• Reciprocals: $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.
• Example: in a 3-4-5 triangle, for the angle opposite the side 3, $\displaystyle \sin\theta = \frac{3}{5}$, $\displaystyle \cos\theta = \frac{4}{5}$, $\displaystyle \tan\theta = \frac{3}{4}$.