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

Verified step by step guidance

1. Recall the reciprocal identity relating sine and cosecant: \(\csc \theta = \frac{1}{\sin \theta}\).
2. Given \(\csc \theta = 1.25\), rewrite this as \(1.25 = \frac{1}{\sin \theta}\).
3. To find \(\sin \theta\), take the reciprocal of both sides: \(\sin \theta = \frac{1}{1.25}\).
4. Simplify the fraction \(\frac{1}{1.25}\) by expressing 1.25 as a fraction: \(1.25 = \frac{5}{4}\), so \(\sin \theta = \frac{1}{\frac{5}{4}}\).
5. Use the property of dividing by a fraction: \(\sin \theta = 1 \times \frac{4}{5} = \frac{4}{5}\). This fraction is already rationalized.

Key Concepts

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as sine and cosecant. Specifically, sin θ = 1 / csc θ. Understanding this allows you to find one function value when given its reciprocal.

Rationalizing Denominators

Rationalizing denominators involves eliminating any irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression, making the expression simpler and more standard.

Evaluating Trigonometric Functions from Given Values

When given a trigonometric function value, such as csc θ, you can find related functions like sin θ by applying identities and performing algebraic manipulations. This process often requires careful substitution and simplification.