Reciprocal Trigonometric Ratios

Property Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocal ratios of sine, cosine, and tangent. $$ \operatorname{csc} A = \frac{1}{\operatorname{sin} A} = \frac{\text{hypotenuse}}{\text{opposite}} \\ \operatorname{sec} A = \frac{1}{\operatorname{cos} A} = \frac{\text{hypotenuse}}{\text{adjacent}} \\ \operatorname{cot} A = \frac{1}{\operatorname{tan} A} = \frac{\text{adjacent}}{\text{opposite}} $$.

Explanation Think of these as the "upside down" versions of SOH CAH TOA. Cosecant flips sine, secant flips cosine, and cotangent flips tangent. If you know the first three ratios, just flip the fraction to find these! It’s like a buy three get three free deal for trigonometric ratios, which makes your life six times easier when solving problems.

Examples If $ \operatorname{sin} A = \frac{3}{5} $ and $ \operatorname{cos} A = \frac{4}{5} $, then their reciprocals are: $$ \operatorname{csc} A = \frac{5}{3} \text{ and } \operatorname{sec} A = \frac{5}{4} $$. If $ \operatorname{tan} A = \frac{3}{4} $, then the cotangent is the reciprocal: $$ \operatorname{cot} A = \frac{4}{3} $$. In a triangle where $ \operatorname{sin} B = \frac{5}{13} $, the cosecant is: $$ \operatorname{csc} B = \frac{13}{5} $$.