Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1.
sin θ , given that csc θ = √24/3

Verified step by step guidance

1. Recall the reciprocal identity relating sine and cosecant: \(\sin \theta = \frac{1}{\csc \theta}\).
2. Substitute the given value of \(\csc \theta = \frac{\sqrt{24}}{3}\) into the identity: \(\sin \theta = \frac{1}{\frac{\sqrt{24}}{3}}\).
3. Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\sin \theta = \frac{3}{\sqrt{24}}\).
4. Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{24}\): \(\sin \theta = \frac{3 \times \sqrt{24}}{\sqrt{24} \times \sqrt{24}}\).
5. Simplify the denominator using the property \(\sqrt{a} \times \sqrt{a} = a\) and reduce the fraction if possible to express \(\sin \theta\) in simplest form.

Key Concepts

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as sin θ = 1/csc θ. These identities allow you to find one function value when its reciprocal is known, simplifying calculations and problem-solving.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This process makes the expression simpler and more standard in form.

Simplifying Radicals

Simplifying radicals means expressing square roots in their simplest form by factoring out perfect squares. This helps in reducing expressions like √24 to 2√6, making calculations and rationalization easier.