Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. sin θ , given that csc θ = 1.25

Key Concepts

Reciprocal Identities

Reciprocal identities in trigonometry relate the sine, cosine, tangent, and their respective cosecant, secant, and cotangent functions. Specifically, the cosecant function is the reciprocal of the sine function, meaning csc θ = 1/sin θ. This identity allows us to find the sine value when given the cosecant value, which is essential for solving the problem.

Rationalizing Denominators

Rationalizing the denominator is a technique used to eliminate any radical expressions from the denominator of a fraction. In trigonometry, this often involves multiplying the numerator and denominator by a suitable expression to achieve a rational denominator. This process is important for presenting final answers in a standard form, especially when dealing with trigonometric functions.

Function Values

Function values in trigonometry refer to the specific outputs of trigonometric functions for given angles. Understanding how to compute these values, such as sine, cosine, and tangent, is crucial for solving problems. In this case, finding sin θ from csc θ involves applying the reciprocal identity and understanding the relationship between these functions.