Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. csc θ , given that sin θ = ―3/7

Reciprocal identities in trigonometry relate the primary trigonometric functions to their reciprocals. For instance, the cosecant function (csc) is the reciprocal of the sine function (sin), meaning csc θ = 1/sin θ. Understanding these identities is crucial for solving problems that require finding one function value based on another.

Rationalizing the denominator is a process used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will simplify the expression. In trigonometry, this is important for presenting answers in a standard form, especially when dealing with functions that may involve square roots.

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. In this case, knowing that sin θ = -3/7 allows us to find csc θ using the reciprocal identity. Understanding the sine function's properties and its range is essential for interpreting and solving trigonometric equations.