Define the reciprocal identity for sine and cosine. How can these identities be represented mathematically?

Reciprocal identities are fundamental relationships in trigonometry that express one trigonometric function as the reciprocal of another. Specifically for sine and cosine, the reciprocal identities involve the functions **cosecant** and **secant**, respectively. These identities are essential for simplifying trigonometric expressions and solving equations. ### Reciprocal Identities for Sine and Cosine 1. **Cosecant as the Reciprocal of Sine** The cosecant function is the reciprocal of the sine function. Mathematically, this relationship is expressed as: \[ \csc(\theta) = \frac{1}{\sin(\theta)} \] - **Domain Consideration:** Since division by zero is undefined, \(\csc(\theta)\) is undefined wherever \(\sin(\theta) = 0\). This occurs at integer multiples of \(\pi\) (i.e., \(\theta = n\pi\), where \(n\) is an integer). 2. **Secant as the Reciprocal of Cosine** The secant function is the reciprocal of the cosine function. This relationship is mathematically represented as: \[ \sec(\theta) = \frac{1}{\cos(\theta)} \] - **Domain Consideration:** Similarly, \(\sec(\theta)\) is undefined wherever \(\cos(\theta) = 0\). This happens at odd multiples of \(\frac{\pi}{2}\) (i.e., \(\theta = \frac{(2n + 1)\pi}{2}\), where \(n\) is an integer). ### Summary of Reciprocal Identities | Function | Reciprocal Identity | |----------|----------------------------| | \(\sin(\theta)\) | \(\csc(\theta) = \dfrac{1}{\sin(\theta)}\) | | \(\cos(\theta)\) | \(\sec(\theta) = \dfrac{1}{\cos(\theta)}\) | ### Example Usage Suppose you know that \(\sin(\theta) = \frac{3}{5}\), and you want to find \(\csc(\theta)\): \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \] Similarly, if \(\cos(\theta) = \frac{4}{7}\), then: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{4}{7}} = \frac{7}{4} \] These reciprocal identities are particularly useful in various applications, including solving trigonometric equations, integration in calculus, and modeling periodic phenomena in physics and engineering.