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

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Recall the reciprocal identity relating sine and cosecant: \(\csc \theta = \frac{1}{\sin \theta}\).

Given \(\csc \theta = 1.25\), rewrite this as \(1.25 = \frac{1}{\sin \theta}\).

To find \(\sin \theta\), take the reciprocal of both sides: \(\sin \theta = \frac{1}{1.25}\).

Simplify the fraction \(\frac{1}{1.25}\) by expressing 1.25 as a fraction: \(1.25 = \frac{5}{4}\), so \(\sin \theta = \frac{1}{\frac{5}{4}}\).

Use the property of dividing by a fraction: \(\sin \theta = 1 \times \frac{4}{5} = \frac{4}{5}\). This fraction is already rationalized.

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주요 개념

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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.

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